What you are talking about doing is an example of an art form known as Machinama. It is where you use a video game engine to play out the actions of a script and voice over it to produce content. One widely successful such project is Red vs Blue which is based around the game engines used in Halo. There are also quite a few different examples of machinama based around the World of Warcraft engine.
For your particular objectives, you would want something that makes it easy to setup camera positions and position actors. Gary's Mod, while a bit dated, is one option for such a sand box. There is also the slightly more advanced Source Filmmaker which is based on the same engine (Source) which is what Half Life 2 was based on. It's a bit dated by today's standards, but both are pretty easy to use and might make a good introduction.
Once you get comfortable with those, it would just be a matter of moving to a more up to date game engine that features realistic motion and rendering. Crytek's engines are pretty well respected for their flexibility and I believe they have pretty good support for things like custom camera positions.
If these turn out to be too complicated, using something with multiplayer and an observer mode can work as a camera as long as you can remove the UI.
Video editing itself is unfortunately a crap in/crap out kind of process. You can't magically enhance the footage you get from the game and make it look more realistic. You can do color grading (adjusting white balance, contrast, color balance, etc) to try and make more realistic tones, but if the textures and lighting and motion are inherently unrealistic, you aren't going to make them look significantly more realistic after the fact in post.