Are there security checks on private flights in your own plane?
No, at least not at the Class C and D airports I have been to. There is just a locked gate with an attendant who makes sure you are either flight crew or are boarding a flight. This is to prevent the possibility of a random person walking around on the ramp.

The PIC of the aircraft, of course, may question you ("your hunting rifle is unloaded, right?") or may search your bags or body cavities or whatever he wants to do before he will let you get on. It depends on the situation. For a charter flight normally they will just verbally talk to you to make sure you aren't unknowingly bringing anything dangerous aboard.

When you fly into a country from a different country, everybody has to go through customs. So, what happens is that you all wait around in the lobby of the GA terminal while some dude from the airport examines your customs forms and goes through your luggage. Sometimes they will search the whole plane too if they want to be dicks. In smaller airports, like in a Caribbean island, you will be routed through the same customs lines that commercial passengers use.