Peekskill schools tend to get a bum rap because the test scores are lower than most in the area. This is why prices in town are so low. But I do know people who send their kids to the schools and are very happy. The town just completed a brand new middle school building. There are a lot of outreach programs with local artists in the schools, too. From what I've heard, the system has many of the problems you'll find in any area with a low income population, but it also has a lot of creative programs and great teachers.
The district also has a great group that raises money for the schools and does outreach programs--
Peekskill Education Foundation home
You can pay to send your kids to Hendrick Hudson if you do not live in the district. But they won't guarantee space. I know that Yorktown charges 17K a year, so I imagine HH is about the same.