The fastest way to build strength in handstand pushups also happens to be the easiest; simply spend more time in a handstand. Anywhere from 10 to 20 minutes should be appropriate for most athletes. This is total time including both work and rest, although try not to go overboard with rest times.
Free balancing on the rings is the absolute king and will lead to strength increases over an incredible range of exercises. Next is free balancing on either the floor or parallets. Finally for those who are as yet unable to free balance, wall handstands will also give a nice return for the effort involved.
Another tip that will help to increase your HSPU strength is to simply hold some of your handstands in a bent arm position. This is actually quite a bit of fun and helps to break up the monotony.
Yours in Fitness,
Coach Sommer