Boring but plausible possibility: It doesn't feel like anything except maybe a slight drain on their mental faculties because it's running in the background and doesn't interact with their conscious mind

More interesting variation of that: They don't consciously perceive what's happening in the game but if they zone out they start to daydream about it

Obvious option that's too simple and straightforward to do much with: They play the game themself, internally, instead of running it for an external player

Funniest option: They hook themself up to a monitor and keyboard/controller and use themself as a game console so they can play video games on their own brain

I'm confident that this is something people have figured out how to do in the Overwatch universe, you can run Doom on anything, it's more a question of "what specifically does that amount to" and "how do the omnics feel about this"