How do I break up with my live-in girlfriend?

Tell her that you want her out, offer to give her istance should she choose to move back home, and giver her a reasonable amount of time to do so say...2 weeks. It'll be awkward, but at least she'll have some notice. Also, if you want her out of your life, don't make a huge deal about a puppy. If she wants it, it's the price you pay to get your life back.