The Trojan War began after the Trojan Paris abducted a Greek woman, Helen, whose husband was Menelaus. Despite Paris' role in starting the war, he is mentioned comparatively infrequently in Homer's Iliad.
One of the most useful places to look for a description of Paris is Iliad 3, where he fights a one-on-one duel with Menelaus. Menelaus would have killed Paris if Aphrodite had not rescued him from the battlefield and whisked him away to Helen's bedroom at Troy, where Helen tells Paris that he is no match for her real husband:
my advice would be to stay here, not fight hand to hand with red-haired Menelaus, nor taunt him rashly, lest his spear conquers you (A.S. Kline translation)
Another of the more telling characterizations of Paris appears at the conclusion of Iliad 6, where Hector and Paris prepare to head out together to the battlefield. Homer, as with many other heroes, describes Paris as "Godlike" and does make Hector say that Paris cannot be faulted for lack of courage. However, Hector goes on to say that "you malinger when it suits, and shun the fight" (Kline translation).
We might also note that Paris is best known as a warrior who fights with a bow, a weapon that allows the fighter to stand farther away from the battle than those who fight with spears.
Even though Homer portrays Paris as a rather cowardly fellow in comparison to Hector or some of the other warriors on the Greek side, the dying Hector does predict (correctly) in Iliad 22 that it will be Paris who ultimately kills Achilles.