This question is interesting, because it seems to focus alone on the positive achievements of Achilles, which is slightly difficult, as he is a character who throughout this text is associated with mighty deeds and with military prowess, but at the same time every deed is also linked to his immaturity and temper and his inability to control his emotions. This is something that is highlighted at the very beginning of the text as a whole, when the narrator says many men died at the hands of Achilles:

Sing, O goddess, the anger of Achilles son of Peleus, that brought countless ills upon the Achaeans. Many a brave soul did it send hurrying down to Hades, and many a hero did it yield a prey to dogs and vultures, for so were the counsels of Jove fulfilled from the day on which the son of Atreus, king of men, and great Achilles, first fell out with one another.

Achilles simply isn't praised for his exploits in this poem, as this quote indicates. His military skill as a warrior is only said to result in "countless ills" for the Achaeans, and the death of "brave souls" and the slaughtering of heroes who were turned into carrion as a result. So, although the poem does testify to his skill and prowess, and these events are highlighted through his killing of Hector, and the way he confronts the river Xanthus, at each point these exploits are reported in such a way that indicates his negative aspects. For example, after defeating Hector in battle, Achilles deliberately desecrates his body in front of the Trojans. In fact, in the text, Hector is compared with Achilles in a way that makes him appear much better and more noble than the somewhat petty figure of Achilles.