You cannot "trash" Game Center, as it supplied by Apple as an original iPhone feature.
But you can "tranquilize" it.
For one thing, stop it by double clicking the home button, finding it on the bottom row (if it's not in the first four items, swipe right-to-left until you see it), holding it down, and then clicking on the minus sign to remove it from the currently running apps.
If it still interrupts you, make sure you're logged out of it. Then stop it as in the last paragraph.