If you ask "how to mod games?" then most people will assume you are interested in making your own mods. "How do I get and apply game mods?", or some variation on the theme, being the better thing here. I will leave it though.
Wii, Wii U doesn't really matter -- the Wii U's Wii mode is effectively its own thing (why you can brick a Wii U's Wii mode and still have the Wii U work) and nobody really kept up with the hack launched mods after the earliest things.
Anyway each hack should have a readme/nfo/whatever with it the details how you will need to apply the hack. Every hack will vary here and the Wii hardware setup, and Wii homebrew/ROM hacking scene, both did things that might complicate it.
Generally one of three things will happen. This is just an overview, do read the included readmes for the given hacks you want to use
1) Conventional old school patch. Here you will be given a patch in one of the various formats suitable for this (bsdiff, xdelta, I heard ppf (playstation patching format) made an appearance for a few hacks). The trouble with this is Wii games are encrypted so you will necessarily be including some data beyond the patch in this so most did not do it. Worse is as time went on the various formats of Wii iso did all sorts of things (first it was scrubbed vs unscrubbed, then various types of scrubbing appeared, then the various packing formats like WBFS started appearing...). Today we have very nice tools like
https://gbatemp.net/threads/new-wii-unscrubber-swiit.511181/ which you can use to work around most of this and get you back to something like a normal ISO.
2) You will be required to present files in some way. For some this can be a matter of extracting the file the game mod needs, patching it using whatever format the modders care to use and inserting it back into the game. Other times they will give you a bunch of programs and a batch file (or standalone program) to do it all for you, though this will tend to mean you need Windows. You might be required to feed it the Wii common key which I can't link/paste here but should not be hard to find, some might even have means to generate it. Wii Scrubber and
https://wit.wiimm.de/ are the main tools I see used to do such things here. If you know what you are doing you can figure out what the batch files are doing and replicate them on another system if you need to.
3) Some kind of hack launched thing. The first example of this exploited an old system menu failure to launch the game with hacks, even on an unhacked system. Said failure was later patched so it went back to conventional methods. More notable however is the Super Smash Brothers stuff. Here it was noted that Smash Brothers had a custom stage option and said custom stage was able to be exploited to load more far reaching game mods. Many Smash Brothers hackers then bundled said game mods with the exploit and thus you can launch many hacks for Smash Brothers with what is essentially an SD card with a custom level and the hack data on it.
