Easiest and best bang for the buck would be:
- Remove governor (FREE) (LINK)
- Aftermarket Header
- Quality high-flow air filter
- Advance Timing 6 degrees
- Flattop Piston
- Rejet Carb
All of the above can be done for about 100-150 dollars, and will give your motor a good jump in power without sacrificing reliability, and without spending big bucks. Unless I was racing in a class that required a 6.5hp clone motor, I would not go any further than this.
From there, power gets a LITTLE bit more expensive, because you will probably want to upgrade your flywheel, buy a new carb, swap out the rod for a billet piece, ect. You have to ask yourself if it is worth it to dump a bunch of money into the motor, or just jump on EBAY and spend 300 dollars on a 13hp Electric start Honda clone. If you do the above said mods to the 13hp motor, you would probably be close to 16 or 17hp.
Here are a couple of good vendors to work with:
Affordable Gokarts
NR Racing
Spend some time on these sites and look at what they have, you will learn a lot.