i had the same problem. it ended up that i wasnt getting compression because the entire thing was pitted and a piece of crap so i ended up buying another bike for the engine. but the battery has nothing to do with the starting. Its just there for the horn, turn signals and that stuff.
First check to make sure you have spark by keeping the spark plug plugged in but grounding it against the outside of the cylinder. I made the mistake of holding it into place and got zapped pretty hard. but at least i figured out i had spark
If there is no spark its probably the plug that needs to be replaced
If there is spark:
I would go out and buy a can of starter fluid and take the airfilter cover off and spray it directly into the air intake tube. if it starts up you know its something with the carb and fuel lines. if it doesnt start, try try again and take it to a local lawn mower fixing shop or auto shop and they can tell you whether or not you are getting enough compression.