I wouldn't say paleo is the cheapest way to eat for someone on a budget, not the way I eat anyway, but it could be done with careful planning. I'm a fan of using the basic tenet's of paleo, but tweaking it as necessary for your own body and training.
Before setting foot in the store I would make a list of everything you want and price check online for a general guideline. Mix and match your items in whatever quantities best suits you and your $70 budget. This way you can maximize your money and know what and how much you can buy at the store.
Here is a list of items to choose from based on what little I know of you-
Weekly Items (not all at once):
Eggs, eggs, and more eggs (did I mention eggs?)
Lean ground beef
Tuna (yuck, I dont eat this but it's another cheap protein)
Canned fruit (in it's own natural juice of course)
Fresh vegetable (spinach!)
Frozen vegetable (broccoli!)
Bi-weekly or Monthly items to factor in:
Frozen fruit (for smoothies, add fresh spinach to them)
Bulk nuts (walnuts!)- no peanuts
Coconut Oil (use in place of butter for cooking eggs, sear chicken breasts, add to smoothies)
Your meals, depending on daily activity, should basically be:
protein, vegetable, and or fruit
protein, vegetable, rice or potato
snacks of fruit and nuts
(boiled eggs on hand always, great for quick snack or a make it a meal)
Don't get hung up on body fat %. Follow a good eating strategy (modified paleo in this case), giving your body the proper fuel to train hard and recover and then see where that puts you after a committed effort. Tweak calories as necessary.