Strengthening your grip is a pretty wide array of topics, truthfully. When we talk about grip strength we talk about a few different things: crushing, supporting and pinching.
Crushing is what you have when you close a gripper. It isn't the same thing as supporting, like holding a barbell. Lastly you have pinching, which is very heavily dependant upon the strength of your thumb. The others... not so much.
Hook gripping has you wrap your hands around the barbell, then overlap your thumb with (typically) your first and middle finger, though there are variations on this. Try it some time, hurts like hell. I suppose you get used to it, but it replaces a strap with your thumb... I've played with it a bit, but in all honesty I don't see the benefit in hooking while deadlifting.
mixed gripping, one hand over and one under, might take some getting used to, but for the vast majority of trainees it is the best possible method to grip more weight without straps. In a dual overhand grip the bar tends to roll. At some point this happens to everyone... however if you hook, there is nowhere for the bar to roll to, as your thumbs are strapping the bar in (did I say OUCH earlier?
) In a mixed grip the bar is stopped from rolling by the opposition of the hand. Simple solution.
The other thing you can do is simply use straps. go without as long as you can, then switch to using straps. If you are not planning on competing in powerlifting, I simply can't see a reason to avoid straps. If you want to train grip, train grip. The DL isn't really going to train your grip.