My recommendation is that you shouldn't actually spend any time doing this because you're going to miss a whole lot of stuff.

For example...

I see nothing in here to handle 1099 employees.

I see nothing in here to handle taxes right down to the city level.

And if you do write one, the testing will take a whole lot longer than you could ever imagine.

My bottom line recommendation is to use some form of 3rd party software that is well established and very well proven. If you want control over the software, the buy "Quick Books", will also allow for imports and exports if needed. If you don't care about absolute control, the sign up for a Web Service that will do it.

Payroll is one of the most difficult things there is to get right. Considering that you're asking how to design the database, I'll suggest that you're not going to get it right. Buy a 3rd party solution.