Building your own Desktop PC is the most affordable way to achieve a quality gaming experience.
The most expensive part will be the video card, and that's the most important thing for gaming.
8GB of RAM should be enough for what you're trying to do.
For the rest of the computer you shouldn't have to use top-notch parts, as they aren't likely to impact your gameplay much.
For all parts you buy I recommending looking up reviews and specifically look at the bad reviews to ensure that none of them have faults / common problems.
If you want a portable gaming PC, for example because of living at university or often going places to game with friends, there are options for cases that can do this. I use a Corsair Graphite 380T for my case and have had few issues moving it from place to place. There are carry covers for monitors as well if you want that.