Since I'm car less (as opposed to car free), I have to do all my shopping on a bike.
I've found a back pak works great.
Get one large enough to shoehorn a 12 pak of double thick TP and you'll have the right size!
Plan your trips so you carry some heavy items with some lighter weight items. You don't want a pak full of canned goods. It gets too heavy and strains the pak. (I came home once with 53 lbs. of stuff in the pak and hanging off the bars. WAY TOO MUCH! I now have a 30 lb. limit and try to keep it closer to 25.
Items like bags of chips etc., can be hung off the handle bars.
Heavier items like a 6 pak of pop need to be "strung". Hang them off to the side and they start swinging, messing up the steering. Hanging off both sides and they start getting into harmonics and make it virtually impossible to steer. I "string" them between my thumb shifter bosses. (not the levers)
edit:
Also anticipate your needs. If you happen to be out for whatever reason, stop by the store on your way home to pick up a couple "lighter" items. I always keep a "mental" grocery list of what I can carry without the pak. Since I've recently become a chips & salsa freak, I'll often pick up chips. It seems that when I get chips with other items, they tend to get broken up smaller.