My brother in law has been living in our garage since November 2009, he agreed to pay us $300.00 a month if we let him and his two dogs live with us. He has still to this day has not paid us a dime, and his dogs have destroyed our backyard. Can I evict him without causing legal problems for myself?

You might have a local ordance against people living in garages. Yes, this would pose a problem for you BUT if you really want him out you could get the county or city involved and cite you for having him live there illegally which would mean that he HAS to move out immediately.

You might want to call up and find out how much of a fine you would get for having him there illegally before going forward.

If you were to take him to court and legally evict him he could pipe up at the hearing that he's living in your garage illegally and the Judge would certainly look at you like you had green and yellow polkadots on your suit.

Well first the fact you never tried to collect any rent, he could argue that in fact there was never any real rent to be paid. But you can of course unless he has a written lease merely just give him notice and hopefully he will just move. I would start with a 30 day notice.
If he does not move, then the issue of filing in court for a formal eviction

Unless you can prove the agreement to pay rent, it may be hard to evict him for non payment. But as noted, you can give him a notice of termination and give him 14-60 days to vacate. Local laws will govern the term. You can check with your local housing court for the rules of the eviction process.