The phrase (as) drunk as Chloe means very drunk.

This is the origin of the phrase, according to the English clergyman and schoolmaster Ebenezer Cobham Brewer (1810-1897) in Dictionary of Phrase and Fable, giving the Derivation, Source, or Origin of Common Phrases, Allusions, and Words that have a Tale to Tell (London: Cassell, Petter, and Galpin, [1870?])—Matthew Prior (1664-1721) was an English poet and diplomat:

Chloë is a lady mentioned often in Prior’s “Poems,” who had a great propensity for strong drinks.

And this is what is said of the relationship between Prior and Chloe in An Antiquarian Ramble in the Streets of London, with Anecdotes of their more celebrated Residents (London: Richard Bentley, 1846), by the English painter, engraver and antiquarian John Thomas Smith (1766-1833), edited by the Scottish author and journalist Charles Mackay (1814-1889):

It is alleged of Prior, the poet, that after having spent the evening with Oxford ¹, Bolingbroke ², Pope ³, and Swift ⁴, he would go and smoke a pipe and drink a bottle of ale with a common soldier and his wife, in Long Acre, before he went to bed. This woman, the soldier’s wife,—some say a cobbler’s and some an ale-keeper’s wife,—was the beauty whom he celebrates under the name of Chloe. One of his poems to her begins thus,—
     “When Chloe’s picture was to Venus shown,
      Surprised, the goddess took it for her own.”
And another ends with the following, of which the English is not very choice, or the rhyme very euphonious,—
     “Then finish, dear Chloe, this pastoral war,
      And let us, like Horace and Lydia, agree;
      For thou art a girl as much brighter than her,
      As he was a poet sublimer than me.”
Pope says, every body knew what a wretch this woman was, and adds, on another occasion, “Prior was not a right good man; he used to bury himself for whole days and nights together with this poor mean creature, and often drank hard.”

¹ Robert Harley (1661-1724), 1^st Earl of Oxford and Earl Mortimer, was an English statesman.
² Henry St John (1678-1751), 1^st Viscount Bolingbroke, was an English politician, government official and political philosopher.
³ Alexander Pope (1688-1744) was an English poet.
⁴ Jonathan Swift (1667-1745) was an Irish satirist, poet and Anglican cleric.

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

1-: From Joey’s Christening. A Song in Low Life, published in The New Vocal Enchantress: Containing an Elegant Selection of the Newest Songs Lately Sung at the Theatres Royal (London: Printed for C. Stalker, 1788):

They swill’d gin-hot
Until, blind drunk as Chloe,
At twelve, all bundled from
The christ’ning of young Joey.

2-: From Life (Dublin: Printed for P. Wogan, H. Colbert, W. Porter, etc., 1801), a comedy by the English playwright Frederick Reynolds (1764-1841):

I say, guardy, there’s no fear of his finding me out—for ecod! he’s as drunk as Chloe.

3-: From The Englishman in Paris; A Satirical Novel. With Sketches of the most remarkable Characters that have recently visited that celebrated Capital (London: Printed for Sherwood, Neely, and Jones, 1819)—author unknown:

Three mad fellows, as drunk as Chloe, charged through the ninety-second regiment. Our brave fellows did not like to kill ’em at first; but enough is as good as a feast; and when they began slashing about right and left, they very properly knocked them on the head, (pour encourager les autres.)

4-: From The Scribbler: A Series of Weekly Essays, on Literary, Critical, Satirical, Moral, and Local Subjects; interspersed with Pieces of Poetry (Montreal, Quebec, Canada) of Thursday 16^th January 1823:

It is at last fixed that the beautiful Miss Sally Fitzmaurice, of St. Lewis ward, is to be joined in wedlock’s bands to Capt. Sansculotte, of Lottville. The courtship it is said, did not exceed five years. The lady’s father is much pleased with the match, and is determined to give the ladies and gemmen, what he terms a damn’d good blow-out, and make them all “as drunk as Chloe.”

5-: From Slang. A Dictionary of the Turf, the Ring, the Chase, the Pit, of Bon-ton, and the Varieties of Life, Forming the Completest and Most Authentic Lexicon Balatronicum ⁵ Hitherto Offered to the Notice of the Sporting World (London: Printed for T. Hughes, 1823), by the British author and journalist Jon Bee (John Badcock – fl. 1810-1830):

‘Drunk, positively;’ too much for a man’s reasoning powers. ‘Drunk as Chloe;’ she must have been an uproarious lass. ‘Drunk as Davy’s sow;’ a heavy swinish departure of the faculties. A thousand other grades of drunkenness might be quoted, but we cite only one more: ‘as drunk as a fiddler’s bitch,’ would imply, that the patient has the buz of music in his ears and will not sit quietly, but danceth about.

⁵ The adjective balatronicum was derived from the Latin noun bălātro/ōnis, which denoted a babbler, hence a jester, a buffoon. This noun was in turn derived from the verb blătĕro, to talk idly or foolishly, to babble, prate.

6-: From the Vermont Statesman (Castleton, Vermont, USA) of Wednesday 26^th July 1826:

Preface to a 4th of July Celebration. July 3d, 8 o’clock, P. M.—Squibs and crackers, by 20 or 30 boys […].
9 o’clock.—Boxing match between half a dozen black men on one side, and half a dozen blackguards on the other. The utmost order and good fellowship prevailed between the parties, who after a few rounds repaired to the “Black Swan,” and there got—as drunk as Chloe.

7-: From Miseries, published in the Gloucester Telegraph (Gloucester, Massachusetts, USA) of Saturday 21^st April 1827:

To see a man, whom you suppose possesses some sober moments, drunk as chloe [sic] in broad day-light.

8-: From Police Incidents, published in The Australian. A Commercial, Political, and Literary Journal (Sydney, New South Wales, Australia) of Friday 26^th June 1829:

A blue eyed maid, we do not pretend to charge the frail one with being a maid-en, per Edward, was brought before their Worships, having been found by the guardians of peace and soberness, in a situation rather unbecoming her sex—nothing more nor less than drunk in the streets at a late hour of the blessed night. The assignee of Jane Williams (for such was the offender’s name) swore on the Holy Evangelists, that Jeanie was in reality, though not a Phœnix of serving women, a tolerably good housemaid. But “evil communications corrupt good manners.” ⁶ One of Jeanie’s shipmates, on the voyage out from her native country, had paid her a visit, and Jeanie consequently, with “lowly suit and plaintive ditty,” ⁷ requested leave of absence for half an hour only, and by all that was good and gracious, no longer. So having obtained leave of her mistress, away trotted Miss Jane Williams with her crony. Hour after hour expired, but no tidings came of the woman, till news reached her mistress that Jeanie, oh horrible reverse, was in the watch-house, having been picked up as aforesaid, drunk as Chloe.

⁶ Evil communications corrupt good manners occurs in the first Epistle of Paul to the Church at Corinth, 15:33. It is described, by commentators on the New Testament and by writers on the Greek drama, as a quotation in the first place from the lost Thaïs of the Greek playwright Menander (c. 342-292 BC), and it is further stated that it was there borrowed from some play, no longer extant, of the Greek playwright Euripides (480-c. 406 BC).
⁷ With lowly Suit and plaintive Ditty is the title of a song from No Song, No Supper, an opera with music by the English composer Stephen Storace (1762-1796) to a libretto by the English painter and playwright Prince Hoare (1755-1834); this opera premiered in London in 1790.