To check how many kills a character has accumulated, go to the main menu and click "status". Then hover over which character's status you'd like to check. The kills will be showcased in the upper right corner.
There are three achievements for killing enemies with Marine, one for Karine, and one for Hirado. If you are having troubles with the "500 kills", defend with every character except for the one that you need the kills with. You can easily farm on a weak enemy.