From what you are saying It could be a few things.
First, have you got a good spark at the plug when the engine is warm?
And good compression? Do you have any air intake leaks? and if all that is OK, is, on the carburetor, the choke plunger rubber in good condition and is the plunger closing properly? give the cable plenty of free play.
Check and set the float level,(have you got a manual?) check if the float is not leaking, and that the tab on the float shuts the needle valve properly.
Sometimes symptoms like that could possible be a crank case or seal leaking, maybe letting in air.
Hope this helps.