The classic biography is of course James Boswell, The Life of Samuel Johnson, LL.D., better known as "Boswell's Life of Johnson," "Boswell's Life," or even just "Boswell." The scholarly edition is in six volumes from Clarendon Press, originally edited by George Birkbeck Hill in 1897 and revised by Lawrence F. Powell from 1934 to 1964: the first four volumes are the Life proper; the fifth is the Journal of a Tour to the Hebrides with Samuel Johnson; the sixth is a massive and comprehensive index to the lot. (Powell reproduces Hill's edition page-by-page, although he introduces many corrections and adds many new notes, most in appendices.) The full six-volume edition, though, is out of the reach of most casual readers; for them, the one-volume Oxford World's Classic, edited by R. W. Chapman, will do the trick. There are a number of abridgments, and any of them will serve, I suppose; none, though, wins my seal of approval. (I'm pissed off at the Penguin edition for excluding the three-year-old Johnson's poem on "good master Duck.")
Boswell is himself the subject of mountains of scholarship, not least on just how reliable he is as a biographer. Donald J. Greene is the leading Boswell-basher, in articles like "The Logia of Samuel Johnson," The Age of Johnson, 3 (1990), 1-33 (scholarly), and "The World's Worst Biography," The American Scholar, 62:3 (Summer 1993), 365-82 (less scholarly).
Boswell is hardly the last word on Johnsonian biography — and in fact, he's not the first, either. Two of his contemporaries produced important rival biographies before Boswell: Sir John Hawkins's Life appeared as the first volume of Hawkins's 1787 edition of Johnson's Works. There is no modern edition of the complete text, but Bertram Davis did both a handy abridgment (New York: McMillan, 1961) and a critical study (Johnson Before Boswell: A Study of Sir John Hawkins' Life of Samuel Johnson [New Haven: Yale Univ. Press, 1960]). There's also Hester Thrale Piozzi's Anecdotes of the Late Samuel Johnson, which appeared in 1786, and can be read in Hill's Johnsonian Miscellanies among other places. And a collection of minor early biographies was published by O M Brack, Jr., and Robert E. Kelley as The Early Biographies of Samuel Johnson (Iowa City: Univ. of Iowa Press, 1974).
Modern biographies abound. Walter Jackson Bate's Samuel Johnson (New York: Harcourt, Brace, Jovanovich, 1977) is one of the best. James Clifford covers the early and middle years in two volumes, Young Sam Johnson (New York: McGraw-Hill, 1955) and Dictionary Johnson (New York: McGraw-Hill, 1975). The others are a mixed bag. I'm fond of Robert DeMaria's Samuel Johnson: A Critical Biography, but wouldn't recommend it to newcomers; it's more critical than biographical, and assumes you already know much of the life.
A great mountain of disorganized and undigested biographical information can be had in Johnsonian Gleanings, ed. Aleyn Lyell Reade, 11 vols. (London: Francis, 1909-52).
This is part of a Guide to Samuel Johnson by Jack Lynch. Comments are welcome.