Can a PC wizard become a Blood Mage at paragon path, and later become an archlich at epic path? Since undead don't have blood, is it possible for a lich to keep blood mage path features?
Nowhere (other than vaguely, in the flavor) does it say that you actually need to have blood in order to be a blood mage or use your blood mage powers.
This may seem odd, but that's just how D&D4e works - mechanics (e.g, powers) do what they say they do, nothing more or less. The flavor of a given power can be entirely disjointed from its actual behavior, and this is fine! You are expected to either come up with your own explanation for how the mechanics make sense given the flavor (maybe when you become an archlich, you're going to turn into something closer to a (still blood-filled) zombie than a dry, dusty skeleton), or to come up with your own alternate flavor and say that the mechanics represent it (and you can do this even if the existing flavor works fine).
Note that a) the blood mage paragon path does not exclude shardminds, warforged or other non-blood-bearing PC races, and b) the blood mage has many powers that interact with the Bloodied state, and continue to do so even when it's applied to bloodless monsters like golems or whatever.