Lets say you want to have a big battle, 100 orcs, 50 elves, 50 human soldiers, etc. Draw rectangles on square paper for each group, 1 square = 50 HP. When the orcs attack the elves roll d20 with attack bonus and AC as though it were a one on one fight, but roll a d10 for damage. For each group I write a list of how much damage it does depending on the d10 roll. So for the orcs maybe 1-5 does 50 damage, 6-8 does 100 damage, 9-10 does 150 damage. For humans 1-6 does 50 damage, 7-10 does 100 damage. In my game, non PC combat mostly works as a countdown and prompt for battlefield combat scenarios. e.g for a few rounds the shield wall holds and the PCs are doing area and ranged attacks against a swarm, then the orcs break through and the PCs have to take down the biggest, baddest one. The moment the big bad breaks through is when the soldiers are at half health. Anyone else do battles with 100s of creatures in D&D?

I've been in battles as a PC.

The DM narrates most of the battle, with just little vignettes showing the PCs doing something important during the battle.

No point in showing PCs fighting in a front line of an army. One fireball from a PC will take out a good chunk of a coherent unit of enemies, so don't bother trying to include them in normal battles as normal troops.

Matt Colville has rules for this in his Strongholds and Followers book. You create unit(s) for each side, and let the players control their side and the DM controls the other side. Each time a player takes a turn, or each time a monster gets a turn, one unit from that side can make an attack against another unit.

If the players win the battle before the unit combat is decided (which will happen most of the time, since units only inflict damage on each other occasionally), their units are assumed to ultimately win, and if they lose, their units ultimately lose.