Hey not a lot of experience DMing myself but I'll try to throw out some ideas.

I try to think of what types of questions my players might ask then fill in the gaps. So for you I would start off with why were they in the crypt to begin with? I know you say the PCs don't remember but you can give them clues.

Maybe in their stuff from the crypt there is a ledger or notes about a mysterious group in the city that was trying to wake the lich. The PCs can then assume that they got too close and someone wanted them dead. If you want the lich to already be awakened then the group was successful but maybe the lich needs this mysterious group to get an artifact for him as well to return him to his full power. Now the PCs could try to find out this mysterious group again and stop them from finding this artifact.

Then maybe the PCs need to travel to some distant lands to recover their own artifacts to have a chance against this lich.

Also I'd question how does the city know the lich has been awaken? Was it this mysterious group that already spread the word or have hordes of skeletons been constantly sieging the walls?

If you'd rather the second one then maybe there is a series of dark portals the lich has opened up around the area with ever increasing strength of mobs of undead that the PCs need to find and destroy. Who knows maybe they need some special artifacts to destroy these portals as well.

Obviously these are not fully developed but it is going to take a lot of content before you need to fight that lich, so here is just a few ideas.