The phrase to chase the dragon means: to take heroin by heating it and inhaling the fumes: the heroin is placed on tin foil and lit with a taper, the fumes forming a pattern on the foil that is said to resemble the tail of a dragon.
—Cf. also the phrase to tickle the dragon(’s tail), which means: to undertake a dangerous or hazardous operation or activity.

The phrase to chase the dragon apparently originated:
– as a translation of Cantonese slang chui lung, meaning dragon chasing; ¹
– in the 1950s;
– in East Asia, more specifically in Hong Kong, which was then a British dependency;
– in connection with the systematic production and distribution of narcotic drugs organised as a government monopoly by the Chinese People’s Republic.

¹ According to The Politics of Narcotic Drugs: A Survey (London: Routledge, 2011), edited by Julia Buxton, the phrase is a translation of Cantonese slang chui lung, meaning dragon chasing—cf. also, quoted below, the Bulletin on Narcotics (United Nations Office on Drugs and Crime) of Wednesday 1^st January 1958.

These are the earliest occurrences of the phrase to chase the dragon that I have found, in chronological order:

1-: From The Fresno Bee (Fresno, California, USA) of Saturday 21^st August 1954:

Spreading Narcotic Addiction And Red China
From a speech by Harry J. Anslinger, federal narcotics commissioner ²

Spreading narcotic addiction and obtaining funds for political purposes through the sale of heroin and opium is not just the policy of one man in the Communist regime. It is the policy of the entire Communist regime in mainland China.
[…]
The fanatical Communist narcotic traffickers have resorted to the extreme measure of cutting off the ears of those small time sellers who dared reveal the identity of the supplying group. Another method has been to take them into a Communist organization whereby they are sealed off and thereby silenced.
The golden web of the conspirators protecting them in their palatial surroundings has been spun through the funds ultimately obtained from the very persons destroyed by the product—the very persons who pay in money and blood to chase the dragon or shoot AA guns, terms which are now heard throughout the Orient with reference to the use of heroin.

² Harry Jacob Anslinger (1892-1975) served as the first commissioner of the U.S. Treasury Department’s Federal Bureau of Narcotics from 1930 to 1962.

2-: From an article by Robert F. Coll, United Press Staff Correspondent, published in several U.S. newspapers on Thursday 20^th June 1957—for example in the Orlando Evening Star (Orlando, Florida, USA):

Washington (UP)—Federal Narcotics Commr. Harry J. Anslinger said today that Red China is responsible for more than half of the illicit world traffic in opium.
Anslinger, who also heads the United Nation’s commission on narcotic drugs, said the UN has received evidence of “enormous” shipments of opium from the Chinese mainland comprising 65 pct. of the world traffic.
[…]
He said a recently-completed UN survey also uncovered a “very mysterious” new sex stimulant called Khat that is chewed by African natives. It also disclosed a new Hong Kong practice of “chasing the dragon” in inhaling opium vapors, he said.

3-: From the Bulletin on Narcotics (United Nations Office on Drugs and Crime) of Wednesday 1^st January 1958:

Chasing the dragon: The smoking of heroin in Hong Kong
Information transmitted for publication by the Government of the United Kingdom; furnished by the Government of Hong Kong
The expression “chasing the dragon” is now a widely known and established idiom used in narcotic circles in this colony in connexion with the smoking of heroin. In those parts of the world which, like Hong Kong, suffer from a heroin addiction problem, the popular form of absorbing this type of drug is by means of injection. The hypodermic syringe is not a common feature of local heroin traffic, and our addicts on the whole continue to be shy of the needle. In Hong Kong, where heroin has now superseded opium as the main drug of addiction, the traditional form of smoking has been retained by addicts who indulge in the use of this drug.
Narcotic retailers normally sell very crude heroin in small plain or coloured cellophane packets costing H K $0.50, $1 or $2. The average weight of the powder in these packets is approximately 0.032, 0.063 and 0.126 g respectively. The cheapness of this commodity indicates that the heroin being sold in this form is of a crude quality and is therefore unfit for injection purposes. The large majority of addicts in Hong Kong cannot afford to purchase high-grade heroin, and to overcome this economic factor a method known as “chasing the dragon” has been adopted to satisfy their needs.
Fine heroin powder has a tendency to run into a single mass when heat is applied. The outside of this mass then becomes charred and tends to protect the inner portion of the mass from heat, thus preventing complete combustion, but naturally resulting in a certain wastage. For smoking, therefore, crude heroin in rough granular form is preferred by the local heroin addicts and is available in packages as outlined above. With each purchase, the retailers also issue a base powder commonly known as “daai fan”. This is generally one of the barbiturates, usually also in grain form, the purpose of which is to prolong the effects of the heroin and to aid sublimation of the heroin when heat is applied.
The implements required for the purpose of smoking consist of a piece of tinfoil of the size found in the average cigarette packet, a paper tube similar to a drinking straw, some cardboard spills and a small lamp. The tinfoil is first smoothed out with a bone or metal instrument until it attains a high polish. It is then folded in half and the heroin and base powder are placed in the crease about half an inch apart. A cardboard spill is then lit and the flame is applied to the tinfoil directly underneath the base powder which liquefies and mixes with the heroin. The tinfoil is then tilted back and forth whilst the flame is applied directly under the crease; at the same time the smoker directs the paper tube which is held in his mouth over the tinfoil and inhales the fumes emanating from the heated mixture.
To obtain the best effect the paper pipe chases the “smoky tail” of the liquid which is moving back and forth in the tinfoil trough until nothing is left on the tinfoil except a black stain. With the usual local aptitude for a turn of phrase this whole procedure has been called “chui lung” or “chasing the dragon”.
This simple method of consuming heroin dispenses with pipes and other paraphernalia normally utilized in the smoking of opium and heroin pills. As the minute quantities of crude heroin and base powder are cheap, easy to conceal or throw away, the chances of the addict’s being arrested with incriminating evidence in his possession are greatly lessened.
A pharmacological investigation and evaluation of the effects of combined barbiturate and heroin inhalation by addicts who employ this method of smoking heroin has been specially made in connexion with the preparation of this article by Dr. Carl C. Gruhzit, M.D., Ph.D., Lecturer in Pharmacology in the University of Hong Kong.

4-: From the Evening Express (Aberdeen, Aberdeenshire, Scotland) of Friday 3^rd October 1958:

‘CHASING THE DRAGON’

A habit in Hong-Kong known as “chasing the dragon,” is described in a U.N. bulletin on drugs to-day as “a particularly odious form of drug addiction.”
The report said that information given by the Hong-Kong Government showed that the habit was formed by a large majority of the drug addicts in the colony, who could not afford high-grade heroin.
It said that they buy crude heroin and with each purchase get a base powder known as “daai fan,” usually a barbiturate, which prolongs the effects of the heroin and aids sublimation of the drug when heat is applied.
The heroin and the base powder are put in the crease of a piece of tinfoil and heated until the powder liquifies and mixes with the heroin, says the report. Then the fumes are inhaled through a paper tube—and that is known as “chasing the dragon.”
Dr. Carl Gruhzit, of the University of Hong-Kong, says in the report: “The central nervous system repression produced by these drugs may result in marked torpor. From a medical standpoint the mixture represents a great danger to the individual.”

5-: From Dope Facts Stun Colony: 1 in Every 10 Persons In Hong Kong is Addict, by Jack MacBeth, published in The Sun (Vancouver, British Columbia, Canada) of Wednesday 18^th November 1959:

Hong Kong—Nearly one person in every 10 of Hong Kong’s three million people is now a drug addict.
This fact, emerging from a special White Paper on the colony’s narcotic problem, has stunned both the Chinese and the European communities here and that was just what the government wanted it to do. The long, illustrated document held little back from the public.
[…]
It [= the White Paper] also described various ways heroin can be consumed. Most addicts employ a method known as “chasing the dragon,” in which several grains of the drug are mixed with a base powder in a folded piece of tin foil. This is then heated over a taper. The fumes, inhaled through a small tube of bamboo or rolled paper, move up and down the tin foil trough and resemble the tail of the dragon in Chinese mythology.
The use of a narrow tube to inhale the fumes was said to be relatively inefficient and, instead, the smoker often used a match-box cover.

The following photograph and caption are from Dope Facts Stun Colony: 1 in Every 10 Persons In Hong Kong is Addict, published in The Sun (Vancouver, British Columbia, Canada) of Wednesday 18^th November 1959:

ANGUISHED DRAG on smoldering heroin is taken by crazed, young woman in typical scene of degradations of drug addiction in Hong Kong. The British colony has been stunned by special report on addiction and smuggling problems—Hong Kong police photo.

6-: From the Herald Express (Torquay, Devon, England) of Saturday 20^th August 1960:

Drinking-straw holders are disappearing from some small cafes and restaurants in Hong Kong, and proprietors are issuing one straw with each drink.
One explanation is that too many straws have been taken in the past by customers who wanted them for tubes to “chase the dragon” (smoke heroin).

7-: From Dope, Red China’s Secret Weapon, by Lawrence Sullivan, Coordinator of Information for the U.S. House of Representatives, published in The Lima News (Lima, Ohio, USA) of Wednesday 21^st December 1960:

For the first time in human history, the systematic production and distribution of narcotic drugs has become an organized government monopoly in Red China. In 10 years, Mao Tse-tung has built up a virtual world monopoly in opium, heroin and morphine.
[…]
Harry J. Anslinger, chief of the Narcotics Bureau, U. S. Treasury Department, was the U. S. delegate to the special UN conference on the rapidly expanding narcotics trade.
[…]
Everyone connected with the Communist dope traffic is charged with the daily responsibility and duty of recruiting new addicts to “chase the dragon,” as the habit is described throughout the Orient.
Anslinger documented for the UN Commission the case of a young Japanese seaman, Saito, who signed on one of the ships in the Chinese dope fleet from Yokohama.
During his first voyage Saito, who was not an addict, was tied hand and feet and forcibly given heroin injections until he manifest a craving for the drug. He became a confirmed addict.
Youngsters thus forced to “chase the dragon” become ready servants of the Chinese dope mobsters, eager for any criminal assignment or violence which will assure their daily narcotic requirements. By this technique, Communist China has built up a ready army of fiendish criminals in all the principal port cities of the world.