As other answers here have noted, hot-dog as an adjective in the context of the quoted comment means "show-off" or "grandstanding" or "attention-seeking."
As for when that slang sense of the term originated, J.E. Lighter, The Random House Historical Dictionary of American Slang (1997) finds an astonishingly early instance, from 1894:
hot dog n. 1. Orig. Stu[dent slang] a cocky and proficient individual, now usu. a competitive athlete; (also) one who behaves, performs, or dresses in a flashy, conceited or ostentatious manner; a mere show-off.
[First four cited occurrences:] 1894 in Comments of Ety[mology] (Nov. 1995) 19: Two Greeks a "hot dog" freshman sought./The Clothes they found, their favor bought. 1897 in Comments of Ety. (Nov. 1995) 18: "Brown's a hot dog, isn't he?" "Yes, he has so many pants." 1899 Kountz Baxter's Letters 34: A Messe de Mariage seems to be some kind of a wedding march, and a bishop, who is a real hot dog won't issue a certificate unless the band plays the Messe. 1900 D[ialect] N[otes] II 42: {College slang:} hot dog...One very proficient at certain things....A conceited person.
The "Brown's a hot dog" quotation comes from Wrinkle (June 19, 1897), a student publication of the University of Michigan; unfortunately the reference appears in Punch-style isolation as a two-line dialogue joke. The 1894 instance of "Two Greeks a "hot-dog" freshman sought" is supposedly also from an earlier volume of Wrinkle, although I haven't been able to find that volume online.
The Dialect Notes reference is to E.H. Babbitt, College Words and Phrases (1900), which provides four definitions for hot-dog:
hot-dog, n. 1. One very proficient at certain things. 2. A hot sausage. 3. A hard student. 4. A conceited person.
However, just four years earlier, Willard C. Gore, "Student Slang" in The Inlander (January 1896) notes hot-dog as an entirely positive term (used as an adjective):
hot-dog. Good, superior. "He has made some hot dog drawings for ——."
According to Lighter, hot dog in the sense of "frankfurter" also goes back to 1895,but he doesn't see it as the source of hot dog in the "show-off" sense. Hot dog as "frankfurter" has antecedents going back to the 1840s in the form of the word dog scurrilously associated with "sausage" (as an ingredient). In this regard, Lighter cites D. Corcoran, "A Tourist in Trouble," in Pickings from the Portfolio of the Reporter of the New Orleans "Picayune" 152 (1846):
"MEM.—New Orleans is a wery wile, wicious place : they kills men there with Bowie-knives and dogs with pisoned sassengers. They berries the former holesale in the swamp, and retails the latter, tails and all, as sassenger meat. It's a 'orrible state of society!"
And to like effect, Richard Jackson, "Dutch Warbler," in Popular Songs (originally 1864), to the tune of "oh where oh where has my little dog gone?":
Un sasage ish goot, bolonie of course, Oh where oh where can he be?/ Dey makes um mit dog und dey makes em mit horse, I guess dey makes em mit he."
Leading finally to Farmer & Henley, Slang and Its Analogues Past and Present, volume 2 (1890–1891):
Dogs, subs. (university).—1. Sausages; otherwise BAGS OF MYSTERY (q.v.), or CHAMBERS OF HORRORS, (q.v.).
So hot-dog as "show-off" is almost as old as hot-dog as "frankfurter," and it may have an independent origin. In fact, Dave Wilton, in a February 15, 2009, posting on Word Origins suggests that the flashy sense of hot-dog derived from the idiom "putting on the dog":
This usage is probably a variation on the older expression putting on the dog. From Lyman H. Bagg’s 1871 Four Years At Yale:
Dog, style, splurge. To put on dog, is to make a flashy display, to cut a swell.
It quickly moved from this sense of suave sartorial splendor to proficient, accomplished and eventually to its modern association with extreme sports and risky action.