I've been there often. The vast majority of my diet in tough times consists of beans, rice, and a vegan multivitamin. You're not going to get much cheaper than that regardless of whether or not you're vegan. It's not the healthiest ever, but it'll keep you going and it's honestly still better than what a lot of people in the world eat.
I'm not familiar with what you can get at a food bank, but here are my suggestions for things to buy. I'd imagine a food bank would also have these things, though, since they're all cheap and common foods. Note that this is all from the perspective of being ridiculously broke. I don't recommend eating like this if you can avoid it.
1. Dried lentils, split peas, and any kind of bean. Never get anything in a can unless it's somehow cheaper than dried. I cook two pounds of beans at a time and freeze any that I won't get around to eating before they go bad.
2. Rice. The biggest bag you can find is usually the cheapest. Brown is better for you, so I recommend getting that if you find it cheap enough.
3. Stuff to make bread, biscuits, and other breadlike things (flour, oil, baking powder, yeast, etc.). I'm a big fan of the Savvy Abby biscuits
. Remember that you can make almost anything with water instead of milk. Baked goods get way cheaper when you do that. I always use water out of habit now and everything still tastes great. Whole wheat flour is better for you than white, but it also tends to be way more expensive.
4. Vegan Multivitamin. I don't recommend skipping this. You're really missing out on a lot of stuff when you eat on the super cheap.
Spices, frozen veggies, canned tomatoes, and fruit are good, but are all terrible on a dollar to calorie basis. I'd say buy whatever you can afford here. A solid assortment of spices makes this kind of diet a lot easier. You want to avoid eating junk like potato chips and cookies. Sometimes these are a good deal for the calories, but they're also mostly devoid of nutritional value. You're already in rough territory there and there's no reason to make it worse.
One really important thing is to maintain a reasonable, healthy weight. This is just basic math, but it's obviously going to be cheaper to eat 1800 calories of food a day than to eat 2700. Being overweight is a huge waste of money. It also helps to be really short, but I can't give you any advice there.
Tl;dr version: buy the cheapest food for the calorie, take a vitamin, get/stay in shape. And do whatever you can to stop being poor. You can live like this, and tons of people have and will continue to, but it still sucks.