Engine model & type #'s will help us figure out WHICH carb you have.

IF it's the tank mounted carb, replace the diaphragm and blow carb cleaner through the 2 holes under the diaphragm to make sure they're clear.

You can get an IPL from the Briggs site, which should have an exploded view of the carb.

http://www.briggsandstratton.com/maint_repair/manual_and_more/

IMO If it's and old flat head engine, I would pull the head off, clean the combustion chamber head, head gasket, and watch the valves as you pull the rope. To clean the combustion chamber you will need a wire brush, carb. cleaner, and do it with the valves closed and the piston up. rinse with carb. cleaner to get the little bits of carbon out between the piston and bore. Now you can inspect the valves and seats, bore to piston clearance. every thing looks OK? Put it back togather, change the oil, check compression, and clean the spark plug.

IMO Now your ready to tackle any external problems IMO, remember IMO, IMO, IMO.

Also IMO use carb. cleaner for priming in stead of starting fluid.

IMO, AGAIN IMO, I would also do a fuel system clean up, all the crap in the tank and carb bowl will be finding it'w way to the carb small passages. All of this is routine preventive maintenance on old Flat head engines they carbon up and to keep them running good IMO this has to be done every few years.