Levi's first appearance in the manga is in a side chapter (9.5). What's meant by "side chapter" is that it deviates away from the focus on the main trio to focus briefly on a Survey Corps expedition. From a personal standpoint, I adore this chapter and I think it's a fantastic crash-course of an introduction to this character. Much like Levi's character as a whole, you start out by seeing the surface (that he's kind of a prick) but end up seeing there's a lot more beneath that. The anime equivalent is episode 9, though Levi actually first appears in episode 4 (for like five whole seconds, that hardly even counts).

Before the events in Trost, as mentioned above, a Survey Corps expedition takes place. We first meet Levi (as well as Erwin, Hange, and Petra) as the group prepares to depart. Eren, Mikasa, and Armin watch, and Levi is pointed out as humanity's best soldier, much to his annoyance. As they enter Wall Maria territory, the inherent danger of Survey Corps expeditions is mentioned, with their typical losses exceeding 35%.

This is then immediately illustrated as we flash forward to a soldier hanging halfway out of a titan's mouth. The man, still alive and struggling, stabs at the titan in vain while asserting that humanity will win in the end, and that Levi will take the titan out. Sure enough, Levi then shows up and does indeed kill that titan. We lose sight of the soldier that was in its mouth and the focus shifts to Levi, who is surveying the number of remaining titans in the immediate area. Petra then appears (for the first time) and informs him she's gathered reinforcements. Levi orders her to take care of the fallen soldier(s) and have the rest deal with the titan on their right. He plans to deal with the two on the left by himself. After taking the first one out with ease, he moves on to the second, opting this time to take out its eyes and then takes a moment to chill on its head like it's no big deal. After all, he needs to put on a new set of blades, considering the last ones are lodged in some giant eyeballs, and that's just nasty. We get another idea of how charming he is before he puts that poor bastard out of its misery. It's at this point that we first see his cleanliness fixation—the hell of it is that titan blood evaporates after a short time, but Levi ain't got time for that, so he wipes it all off immediately anyway.

Back on the ground, Levi finds Petra again, who is trying to help the soldier from before. Unfortunately, he is clearly dying, and Petra informs Levi she can't stop the bleeding. The man then beckons Levi over and asks if his death will be in vain. It's here that we get our first look beyond the somewhat prickly surface of this character. The dying man offers his hand to Levi, and it's covered in his blood. Remember that only minutes prior, Levi was fastidiously wiping titan gore from his hand and blades—it's fair to say he's completely grossed out by it. Despite this, with hardly a moment's hesitation, he takes his hand. He then offers some encouraging words, and then promises to eradicate the titans. After he is finished talking, Petra tells Levi the man has passed away. She notes that he appears to have died peacefully, and Petra tells Levi the man must have heard everything he had to say.

At this point, Erwin shows up and informs Levi that they are retreating. Levi, not yet aware of Erwin's reasoning, is less than pleased, and asks Erwin if his men died for no reason. Erwin then tells him that there's been a breach. The chapter ends here, and the story continues back in Trost. Levi doesn't show up again for several more chapters/episodes as the expedition group is on its way back to Trost.