My gym doesn't allow chalk either, but I've been using it there for years. Just get a big zip lock baggie and keep it in there. Chalk up inside the baggie, inside your gym bag, that way it doesn't go all over the place. You don't need a ton of chalk on your hands to the point that its falling all over the floor for the chalk to work. Take an extra towel with you, wet it and wipe down the bar and plates when your done. Usually if you do this stuff they won't hassle you. It's the jakasses that chalk up and clap the chalk all over the place while screaming like gorillas that cause the problems.
As far as grip work:
Barbell holds (Load the bar heavy, pins just before lockout, pick it up and hold it for as long as you can)
Pinch flips
Plate holds (stack 3 or 4 10lb plates side by side and pick the up and hold them, add plates as you get stronger)
High rep double overhand rack pulls (good for traps too)
Farmers walks (HEAVY dumbells)
Any fat bar work
If you do some of that stuff without chalk, then use chalk on your deadlifts you'll notice a difference. Stay away from straps IMO, doesn't matter if your back is stronger than your grip if you can't hold on to the bar in competition.