It is pretty much the same as it is on any other system, except here you have things in a DVD format and need whatever tools you care to use that handle the format and can insert files back into it. Some Square (Enix) games for both the PS1 and PS2 hide things outside the standard DVD format (usually audio) so you will get to handle the raw reads to those locations but hey. Most tools that will be nice to use here will probably be paid software (ultraiso, magic iso and whatnot), though you will probably have fewer formats to contend with than someone wandering into the PS1 (would not be surprised if most were plain iso or bin+cue for the PS2 as standards were sort of a thing by this point).
Beyond that you still get to figure out whatever text encoding the devs dreamt up, or half implemented an existing take on ( http://rikai.com/library/kanjitables/kanji_codes.euc.shtml http://rikai.com/library/kanjitables/kanji_codes.sjis.shtml
for the main Japanese two). This means all the usual relative search ( https://www.romhacking.net/utilities/513/
), editing/corruption, inference, statistical models, tracing and whatnot. Being a file system based thing you have the bonus of file names, extensions, file sizes, directory names and elimination based upon all those, as well as file swapping (many times you can swap files around and as they are generally going to use the same formatting, setup and whatnot you will often see the games still work, just with the files swapped -- quite nice for music hacks and "play as other character"*). For some known formats (albeit for more than the PS2) https://wiki.multimedia.cx/index.php?title=Category:Game_Formatshttp://wiki.xentax.com/index.php/Game_File_Format_Central
Whatever pointers are used (file level or possibly memory level compared to more memory bus/banks of older cartridge based things)
Whatever markup/formatting the devs did for that game (things don't just appear as bold, or stylised, or having a given character portrait)
If you want extra characters then probably learning at least graphics editing enough to edit some existing ones, and maybe some code if you absolutely insist on adding things or want to do something like an 8 bit conversion (should not be as necessary as some other things but memory issues are always a worry on any system).
Whatever oddities the devs did for that game -- text as graphics, some weird formatting, big boy scripting, compression (rare for the PS2 as text is usually a far lesser concern than multimedia -- 4 megs for an entire game script is ridiculously huge, for an audio track with the codecs at the time that is pretty decent, but can be slightly easier on bandwidth** so not unknown) or anything else that joins a list of things that make ROM hacking fun.
I am not sure what we are suggesting for a PS2 debugger these days, though I assume as the main PS2 emulator choice is PCSX2 or not much else then yeah.
Also any source code or hardware documentation that goes with that is probably going to be the main hardware reference, though there are some other documents out there that will give an idea of memory layouts and whatnot. Usually it is whatever is leaked from the official developers (the PS2 was very much in the C programming world so was the start of things not being as detailed as maybe the 16 bit stuff), whatever homebrew developers have (think it was mostly all called homebrew by this point rather than PD/public domain like was common on the PS1, N64 and earlier), and whatever emulator authors have made or generally created as part of emulator source code. Though if you know the CPU you can usually find a basic reference for that elsewhere and start going with that for some aspects. Also not sure what the suggested static disassembler might be if you are going to be heading down that path.
*with the PS2 mostly being a 3d system this is even more likely to work where the random village NPC probably does not have a full fight, movement, action and whatnot animation happening for a 2d sprite sheet, a 3d model though can happily dance to the tune of different animations.
**if you are trying to read say a video file whilst also loading a text file then 500KB of compressed file you sort out later might make life easier than 1 meg of raw uncompressed file you get to use right away. I don't know what the PS2 DVD read speed is offhand but it is not a glorious 16x or anything fun.http://www.romhacking.net/forum/index.php/topic,14708.0.html
is for the GBA and DS mostly but the DS is a file system based console so covers some of the general approaches there that might be missing with the NES focused stuff you have previously seemingly found, as well as statistical (RSTLNE/scrabble distribution, vowels in every word, space every 8 or so characters at most...) and other such things reference above.