Ok, thanks for getting my head on a bit straighter... It sounds like your carb might have a minor problem, since choking it should supply enough cold start encouragement to get it running. Check the carb boot and gasket seal for cracking or leaking between the carb and the cylinder head. You can do this while the motor is idling using a can of carb spray, wd-40 or whatever ya got handy.
Since I don't know what your choke setup looks like (is it cable or lever operated?), I'd look over the service manual and make sure the choke is adjusted right and working, then if that didn't fix anything I'd take the carb off and take it apart for inspection, cleaning and replace all of the soft rubber parts in it. Someone may have had your carb apart before and left something (the float level?) out of adjustment or lost an o-ring or whatever...
Worse case someone might have put a china rebuild kit in the carb, or swapped the OEM carb for a china one...? Let us know how we can help.