A quick and basic check list, and assuming battery is charged and starter is turning the engine OK, and there's fresh gas in the tank (when was the last time it was run, gas goes off after a while):
Is there compression?
If not check:
Valve clearances incorrect. (Too tight)
Valve stuck open
Worn valve guides
Valve timing incorrect (after the backfire it could happen that the timing chain has slipped on the sprockets if the chain was very worn or the tensioner was inoperative.Not very likely. How many miles has it covered?)
Worn piston rings
Worn cylinder
Is there a spark at the plug when you turn the engine over?
If not check:
Fouled or wet spark plug
Faulty CDI unit, pick-up sensor, ignition coil.
Incorrect ignition timing
Open or short circuit in ignition
Petrol getting to the engine?:
No fuel flowing to carburettor:
Blocked fuel filter if there's one.
Blocked fuel tank cap vent hole
Blocked fuel tap
Faulty carburettor float valve
Blocked fuel pipe
Blocked fuel line or carburettor jets
Is there petrol inside the carb float chamber? Check by opening drain screw on carburettor float chamber and see if fuel flows out.
If you have these basic requirements, i.e. Compression, spark and fuel it should at least try to start. The ignition timing on these engines is fixed and can not go out of adjustment. The CDI ignition controller can fail, but if you're getting a spark that should be OK.