Should I make my private class methods static?

If the methods don't access any of the type's state then they should be static.

Static method calls provide a performance gain over instance methods and the presence of a static method tells future readers of your code that calling this method will create no side effects in the the state of the current instance of the type.

The performance gain of a static method comes from the fact that the compiler does not have to emit callvirt instructions to call a static method. The callvirt instruction is handy for instance calls in that it does a null check prior to calling the method. However, when you call a static methods there is no need to check for null so the compiler is free to emit the faster call instruction which doesn't check for null.

As Dan Diplo says, if they can be made static, they should be made static. This is only true for private static methods (which is what you asked about). For public methods, however, they will just confuse users of your class. Public methods should be made static only if they will be used without an instance of the class by callers, the performance loss of not making them static be damned.

I always think if they can be made static (and the compiler will soon let you know if they can't) then they should be made so. See this duplicate question on SO for further discussion.