The Monster Manual leaves the rules about the Lich's new body vague so the GM can decide what happens.
Personally, here is how I interpret it. The Lich's phylactery is very similar to the original Magic Jar spell (not the watered down Magic Jar version we have in 3.5e). So, looking at the similarities with magic jar, I would rule that the Lich can either attempt to possess a conscious creature or possess a slain body/inanimate creature. If the Lich possesses a slain body, then over time that body becomes the Lich. If the Lich possesses a conscious body, then I would follow the magic jar rules for possession.
The magic jar rules, of course, give the victim an excellent chance of resisting the Lich through making a saving throw or otherwise getting far enough away from the phylactery.