Your first step is to define what an ideal balance for your game would be. The easiest way to start this is with simple statements. For example, "Character classes should not disadvantage players when fighting other players" or maybe "Character classes will have advantages over some other classes and disadvantages to others (think Rock Paper Scissors)"
You need to define what balance is for your game.
Once you've defined what balance means for your game, then you need to quantify it. Using the all classes are equal example above: "Average Class Kill/Death ratio, for any two given classes, should fall close to 1.0 within an error tolerance. The error tolerance is set at .5 when any new balancing is added and decreases to a minimum of .05."
So now you can identify what is in balance and what is out of balance with some statistics. The error tolerance part is to allow for statistical errors that should become less significant with more activity. If you see the numbers going out of this range, you know there's a balance issue.
All of this requires that you collect, keep and analyze data about everything in the game. You need this data to run against the quantitative balance checks.
When you do find something out of balance, don't fix it until you have a good idea what is causing the problem. Trying to fix things, without knowing the problem will make things worse. This is why you need to track data on as much about your game as you can, and not just the data specific to the balance.