You'll need to complete both Leon and Claire campaigns, then either character also on the Second Run mode. You will need several runs through Resident Evil 2 in order to garner all achievements a minimum of three, though it's doubtful you will gain everything in just a few runs. If you're just digging your heels into the game for the first time, it's best to simply enjoy your first playthrough.
Welcome to the Resident Evil 2 achievement guide. If it's not labeled by THN (The Horror Network), it's NOT one of mine and therefore not up to the quality that I aim to provide. Note: There have been look-a-like copycat guides popping up.