How do I go about filing for a divorce on my own

Depends on where you are located. I would start at your local Family court.

Are their kids involved, is there property to fight over, will the other party just agree to it and sign the paper work, it all makes a differece, if there is nothing to fight over, you can down load paper work off the net ( for a fee) and both sign it and file. And if you both agree you can both hire the same attorney for about 300 to 400 if you shop around, split the cost and move on