There can be 2 things you are looking for, one is to build from the ground up or two is to make a computer run console games. As you can figure the second is the easiest and cheapest, if the first choice is what your looking at there is an awful lot to consider.
First choice:
Build the console - what processor - type of board - programming - etc.
Creating games - Designing - programming (again) - testing
This would be very time consuming and expensive
Second choice:
Buy a small computer (Even a Raspberry PI would work)
Load an emulator on and your basicaly done - There is a bit more to it, but that is what the internet is for.
Good luck on whichever your choice is, it will be worth it!!!
why i ask is because if you go look under ben heck's youtube channel you will see he has already made a portable pi and I think if you are going to make your own it wil be a great place to start, to see what you will need. I would just inprove on the lcd screen that he used.
