... you have a rather steep learning curve ahead of you if you are in that position (soldering is a basic electronics skill, even if high level aspects of it are something people pay for it their own right. For the purposes of this sort of project it is to writing a story what the ability to type is, or what the ability to use a knife to cut a piece of fruit up is to being a chef)
There are three approaches
1) You redo an android or similar tablet (and there are a lot more to the world of tablets than just ARM based android, the GP32 being an old example but something outside that basic world) to be in a form factor that works for you.
2) You remake an existing console into portable form.
3) You do an actual low level remake of a console using either a similar chip (see famiclones and other such things) or a programmable chip (see stuff like the analogue pocket).
All three likely benefit from some considerable 3d modelling and plastics manufacture skills (though 3d printing is doing amazingly for a lot of this). Making consoles portable tends to need some serious electronics skill as you often have to shrink things down (usually an iterative process as people figure out how to chop and reroute more and more aspects for the purposes of the portable device -- some of the stuff done for portable wiis are notable here) and work out how to power it (consoles originally plugged into the wall often need some odd voltages that you will struggle to get from basic batteries thus needing more creative power arrangements, even more so if it is a power hog though there is also the option to make it work in a car or other 12V source). 1) and 3) do combine with stuff like raspberry pi boards where it is largely bolt together and install software emulators as opposed to make board from scratch and program using very high end programming skills (the FPGAs that power the current hotness are some of the hardest things people can program in many regards).
I am not sure where to start that is not going to make this an essay. For the tablet thing you could probably make a nice mount for it or rear shell that works in it (rather than flimsy clip on) in a game pad style rather than hand killer tablet, maybe stick a nice battery on there for the longer sessions as well.
Portable consoles I already mentioned some stuff about circuit tweaking, though I would also include how people do rapid fire controllers, macro controllers, adapt controllers for disabled people and such like as it will be also the basis for how you take signals out of the controller. Some would also look at video adapters and improvements as getting an old console with RF or something out is not going to work well with modern screens.
Low level stuff... start by writing a basic PC emulator really, and then look up FPGA, CPLD, microcontrollers (though some better options for all the others include some basic microcontrollers). In a pinch you might be able to use a more existing development boards or indeed other single board computers perhaps more suited.
I would not necessarily classify it as making but some also would include those that replace screens, video out, audio amps and such with superior options (all consoles are made down to a price and technology has advanced considerably since the 80s, 90s and 2000s such that what would have been super high end is often 10 cents plus shipping today).