Why is my chainsaw blowing gas out the carb?

Remove the spark plug and squirt a little bit of the fuel from the gas tank into the spark plug hole, replace the spark plug and crank it over several times.

If it runs, or tries to run... that tells you that your engine, ignition system and gasoline are good (or atleast good enough to get it started)

Next check your fuel line and the primer ball. Make sure that when you pump the primer ball that you can see it pumping fuel through the fuel lines in ONE direction only. (It's kind of hard to see, you might have to start the test with empty fuel lines).

If that fails, you need a carb kit because that's where the check valves are. (unless your primer ball is cracked, broken, or clogged)

Next check the metering diafram and needle in the carb. If the diafram is hard (not easily flexible) or if the needle valve is stuck shut the engine will not be able to get the fuel that it needs to run.

If the Needle valve is stuck, just clean it and it'll most likely be fine.

If the diaphram is hardened, you need a carb kit.

If all the above is ok and it still doesn't work... check the gasket at the base of the carb to make sure that the posative pressure pulses from the crankcase are able to make it through the little holes into the carb (that's how it pumps fuel into the carb).

That's pretty much all the easy stuff. Keep in mind that a new carb only costs about $28 at most places.