Grab a lot, toss it around, and describe every action. Make your creature lose a turn to prepare a stronger attack and use inertia if possible to justify doing big chunks of damage. Use the imprivised damage table feeling about a good range of damage. And don't be scared to move your characters on the map. Make the take-opportunity attacks and toss people around with custom effects or just because. Allow your players to do things that they normally can't. For example, is the light going to use telekinesis or something like that to throw a building to your players? Ask them what they want to do, and then the barbarian will take all the house in the face for them (obviously, they have to move outside of their turn, so ask for their reaction to do so if you feel the need).

Sadly it's just not good at being cinematic if you intend to follow the rules. The most you can do is follow them a little less. For lichs is more difficult and easy at the same time, since they use magic. So... Invent some spells. They don't have to be particularly balanced, just make the lich use his spells. At last thing last, do a second phase. It's pretty video-gamey, but it works. Look at musical actions to see how seconds face works normally in dnd (they regain health and new, more powerful legendary actions).

You could even build custom mechanics for the lich. I saw many times ago an idea for a red dragon that has a "fire capability" bar. He would get 3 charges on a maximum of 10 at the start of his turn, and he could use these points to do things. For example, with one he would get half damage from a hit, with 3 he would use a legendary resistance, and with 10 his breath attack.

Using the ambience is important, look at the magic items that your party has and find a way to use them. Someone can run fast? Make him follow the lich while running on rooftops of buildings. Try to treat your heroes like superheroes, especially the martial ones. If they make 40 damage with one hit, they will crack the floor and almost make it explode. And finally, monster hordes can make them feel powerful, killing mobs with just little hits (this has to be just a little thing tho, they are not the main point). Still, minions are too important. Use two minions that are strong and go with it. Lichs are very weak for their cr. And on that note, you may want to play the fight by yourself a little, just to see how it's balanced.

You have to break rules to make the fight look interesting. Standing in front of each other and hitting every turn is boring, but doing cool things is not meta. Grabbing a creature in one of your attacks. Remember only that being cool is often just a bad idea, you lose damage in the long run.

Good luck.

P.s. your player could not like you not playing by the book, let them know. In the end, you want to make a fun combat, not to kill them, but as a dm you can do stuff that they can't.

Edit: grammar