You can try running the carb cleaner of your choice through the gas tank (seafoam, etc.)  Or take a screwdriver and tap on the bowl with the plastic handle to free the float if it is sticking.  Those are purported to fix the problem you describe sometimes.

Having done it a time or nine now, I would suggest removing the bowl and peering around in there with a mirror.  Maybe even spray some cleaner up in the carb.  Make sure to contain it with rags as it runs back out, or it'll screw up your paint.  Pulling the carb in its entirety wasn't as much fun as I anticipated the first time, so I don't do it any more often than I have to.  In fact, I have yet to remove the carb on my bike.  I pulled the one off the parts bike and never tried to reinstall it.  The way others describe it, though, I probably just wasn't holding my mouth right as I did it.  They make it sound easy.

You got it.  That's what I've been doing when changing jets and even just snooping.  It works like a charm.  Pulling the carb isn't major surgery, but it was just enough of a PITA that I won't do it until I really need to.  I just use a little compact mirror of Kay's to look around up in there.

EDIT:  On mine, I have to push back on the battery box to get the bowl off and on, but it'll go.  

ANOTHER EDIT:  Mine is a stock carb.  Don't know exactly what you'll be fighting with.

Thanks for all the advice.  As it turns out, I did not need to remove the carb.  

I tried spraying carb cleaner in the vents, but I have one of Lancer's Mikuni carb so it just sprayed right out of the dial-a-jet.  I then used Sea Foam.  It took about one day of sitting, but now the bike is running as smooth as ever with no oveflow problems.  Of course, there was a lot of banging on the carb with a screwdriver handle.  I'm not sure exactly what did the job, but it looks like something worked.