The phrase alive and kicking means:
– of someone: active and in good health;
– of something: prevalent and very active.

(cf. also history of the phrase ‘alive and well (and living in ——)’)

The phrase alive and kicking is self-explanatory, so that the reason I have transcribed the texts in which it was first used is that they provide some insights into early-19^th-century mentalities.

The earliest instance of alive and kicking that I have found is from a paragraph published in The Royal Cornwall Gazette, Falmouth Packet and Plymouth Journal (Truro, Cornwall) of Saturday 11^th April 1807—the last two occurrences of May in the published text are in italics, in order to emphasise the pun on this word:

Some weeks since, Mary, the wife of Ralph May, of the parish of Gwinear, was brought to bed of two boys and a girl, who, with the mother, are all in good health. On this occasion, the liberality of the neighbouring ladies, gentlemen, and others of inferior rank, does them great honor. This Mary May has had four children within 13 months, still “alive and kicking!” How true then is the idea of Menage and Addison, that May is favourable to population!

The second-earliest instance that I have found is from an “extract of a letter from an officer with Lord Wellington”, published in Relfs Philadelphia Gazette, and Daily Advertiser (Philadelphia, Pennsylvania) of Wednesday 5^th December 1810—this letter gives an account of the Battle of Bussaco, Portugal, between the French forces and the British-Portuguese Army commanded by Arthur Wellesley (1769-1852), 1^st Duke of Wellington:

“I write you this on my cocked hat, not having better accommodation. I have taken the first opportunity after the action, just by way of letting you know that I am alive and kicking.”

The following is from a description of Bartholomew Fair, published in the Caledonian Mercury (Edinburgh, Scotland) of Thursday 9^th September 1813—Bartholomew fair was held annually from 1133 to 1855 at West Smithfield, London; St. Bartholomew, one of the twelve apostles, is regarded as the patron saint of tanners; his feast day is on 24^th August:

Nothing could exceed the wonderful variety which was here collected for the gratification of the noble family of John Bull. The earth, and the sea, the sky, and places never heard of or known in the habitable globe, had been ransacked to produce something worthy of admiration; and certainly a more miraculous assemblage scarce ever before excited so strongly the appetite of curiosity. In one place you saw the truly wonderful lamb, with eleven legs and fourteen heads, all alive and kicking!—in another corner was to be seen the most stupendous elephant, which cracked nuts with his eye-balls, and read the bible with his tail; next came the magnificent Orang Outung, lately imported from the mines of Golconda, whose wonderful performance on the musical glasses astonished and puzzled the vulgar.

On Tuesday 12^th November 1822, The Morning Chronicle (London, England) published this curious paragraph:

Hope for Ireland.—Living beings may, it seems, fall into the worst hands, and still be “alive and kicking.” Those who despair of Ireland, as well as every friend of humanity, will be soothed and comforted by the following record:—“I took,” says Vaillant, in his Tour in Africa, “a large locust off the Cape, opened its belly, and, pulling out its intestines, filled the cavity with cotton; and in that state I fixed it to the bottom of a box with a pin, which passed through its thorax. It remained there for five months, and at the end of that period it still moved both its legs.”