I assume you want to add characters to a font rather than add a whole font to a game.
You do it the same way as you do it on any system.
Find the existing font (and associated encoding and text, or at least have a reasonable understanding of it)
Figure out the graphical and file level format of the font (various options for how fonts work, more later)
Edit (often easier) or add to (often harder but if you have many extra characters then maybe the better choice) as your desires and the demands of the project call for.
Edit table file for your text, type/edit text accordingly.
The PS1 has some shared formats between some developers but nothing resembling a common one shared by many or most games on a system. If it is a popular game (say a Square (Enix) title) then maybe something has already been done.
I imagine the PS1 will still be with bitmap fonts (Windows of a similar timeframe had long been shifting away from that to vector but TTF or bitmap was still a thing there) but you might get lucky and have an in file software defined font rather than the harder limits of the older consoles where it is baked into the program (or hardware) itself, or you might not.
As mentioned most find it easier to edit an existing font than add to it, save in those cases where it is in file software defined and even then editing might still be easier. I don't know what language or symbols you want but presumably not normal English characters. If it is just a few then you can often overwrite some unused punctuation or do other fun things like dual tile encoding and dual characters on tiles (combine ll sort of thing). Otherwise many will opt to use the Japanese game as a base -- Japanese might well have hundreds of characters needed for a game (see Kanji) and thus usually has any number of options here, might also have a bigger script to play with but we can skip that one for now.
Fonts can have many different things associated with them and approaches to them. Generally it will either be everything is a fixed size, everything is a fixed size in the file but there are a list of widths, and everything is sized according to how big it is (makes it a pig to view and edit in a tile viewer) and there is a list of widths somewhere for each character (which you will have to edit if your new character is a different width). I doubt you will have line height values (think j y g f and the like) but I have been surprised in the past.