Since liches are essentially mages who have transformed themselves with negative energies into undead beings and dedicated their lives to all that is evil, I think its fitting to give them a small offensive spell bonus for it. Similarly, mages might also a non-offensive spell bonus when those magic skills are fully implemented.

I think a good idea would be to give lich offensive spells a small bonus, size depending on how close to death they are. For example, maybe adding an additional +1/+2 damage if they are under 45% health, and additional 2/4 damage if they are under 25% health.

If you think this is too much of an advantage, then maybe you can give an inverted but similar penalty when a lich is above 55/75% in health. And in between 45-55% health their casting would be neutral, uneffected by bonus or penalty.

I'm not sure if I'd really like this in practice, cuz it would encourage me to constantly be near death. But i still like the idea.

I agree liches are underbalanced but I don't think this idea would be too practical. For one, I could just cast bless, protection and other benefical spells to lower my HP then gain a bonus in offensive spells. For me, I think I would be spending a lot of playtime trying to lower or raise my HP to a certain point.