just cause its a Tahoe, and it has a decent size stock alt im sure... new Deep Cycle battery SHOULD be just fine, but everyone is gonna tell you to get a new alt and battery.
First thing you should do is upgrade your engine wiring doing the 'Big Three'. If after you install you have dimming, then try upgrading the battery and if that doesn't work then get a HO alt (which would demand you do the big three).