How the fuck do you get a game with two CDs, play through one of them, and just forget that there's a second CD?
Anyway, the way the game works is that there are two "routes". You're going through the same basic concept in each route (you start at the beginning, end on the train ride) but each route takes you through a different path in the city. There's a "Route A" and a "Route B". You play through the game once with one character, and then you play through the second route with the other character.
So if you play through Leon first (Leon Route A) then you start the game with Claire on the other route (Claire B). The opposite thing, Claire A-Leon B, if you start with Claire. The naming convention was because there's some significant differences between Leon A and Leon B routes, and between Leon A vs Claire A. Each character has some paths specific to them, and each route has some parts specific to that route, and there are a few minor differences in one paring which don't show up in the other.
>I need to complete the game 4 fucking times?
Nah, just once through with both characters is enough. Doing things in the opposite way can be fun if you want to see the differences, but you're not really seeing that new of content.
>how does that technology in og resident evil 2 work?
It's been awhile since I played this, but I'm pretty sure that when you put in the other CD, you'll have an option to load a save game from the other character off your memory card. That'll start the Route B for that character.
Note that you need to have played through Route A with the first character. You cannot, say, play through Leon A, then Claire B, then try to use the Claire B save to do a Leon B run.