OK, so these are two things that are hard to get to go together. And something people don't understand is that it's very easy to end up overweight when you're poor. That's because, just trying to get enough to feel reasonably full, you're usually eating a LOT of cheap starches. And telling people they should have the discipline to go hungry all the time is just plain sadistic. The history of dieting shows that while most people can go hungry for a few weeks, very few can go beyond that. When you're poor, you can't substitute high nutritional quality low-fat-and-calorie foods either, and after a while, your body starts desperately craving things because it NEEDS more protein or whatever.
With that said, dieting is one of those areas where watching the small things can make a bigger difference than you think. I realized this when I began checking different brands of 100% whole wheat sandwich bread. A single slice can range from 50 to 120 calories, with the protein content varying from 3 to 5 grams (something you should particularly watch while dieting, you NEED protein). Unfortunately for me, the 50 calorie bread is almost always $2.50 or more a loaf. However, there's another brand that's 60 calories a slice and almost always under $2, and I get it. The cheapest 100% whole wheat bread would save me about $.25 a loaf, but runs about 80 calories a slice, so I consider the trade off worth it. Since I eat about 4 slices of bread most days, I cut out 80 calories a day.
Mayonnaise. I actually use very little of this at a time, but I did compare brands. Normal mayo runs about 110 calories for a tablespoon. I use about a teaspoon or less at a time, but that's still almost 40 calories. I found an olive oil-based mayo that's 40 calories a tablespoon and costs exactly the same as the regular mayo (mayo's one of those things I feel more comfortable buying a "brand" anyway).
Hot dogs. My favorite brand (kosher hot dogs, which I like not only for the taste but because I feel confident in the quality of ingredients) comes in different "fat" contents. The regular hot dogs run about 120 calories each, I think. The reduced fat ones are 90 calories. The fat-free are 40 calories. When these go on sale, I buy up the fat free hot dogs and freeze them for my lunches. I can have two hot dogs on two slices of whole wheat bread with a little mayo for about 225 calories. The regular items would run about 500 calories.
So, paying attention to these little things in my lunch can make 275 calorie a day difference. It take 3500 calories, I think, to lose or gain a pound. This one change ONLY in selecting the item for calories would theoretically take off about a pound in 2 weeks with no other change (assuming you're currently in an equilibrium.) My cheap bean-and-rice lunch soup, with just a little meat added and served over that 60 calorie bread, is, I estimate, about 250 calories per lunch. Cheese in particular is worth watching. I've found mozzarella is usually the lowest calorie inexpensive cheese.
One problem I have to work around is I have IBS and some foods I can't eat. I'll never give you a recipe (that I've personally tested anyway) using banana or any member of the cabbage family (cabbage, broccoli, cauliflower, brussel sprouts, etc.) But something I've found that helps with both budget and diet is making a list of my favorite produce and the calories for them. I can watch for those things to go on sale and stock up. For instance, asparagus is 18 calories for 6 spears (before adding any fats or cheese). Half a cup of mushrooms is 12 calories. Avocados, however, are more like 150-200, possibly higher depending on the size. Sugar is 25 calories a tablespoon, but keep in mind you probably get a lot of hidden sugar in a day if you're eating convenience foods. Added sugar is a good area to eliminate some calories, btw. I've used a measuring teaspoon for my coffee sugar to help me remember.
I also have proteins on my list so I can make a well-thought choice on them. Two large eggs, boiled, are about 150 calories. Scrambled, about 220. Broiled chicken is about 185 calories for 3 ounces, but about 245 for fried. The ground turkey I use is about 175 calories for 3 ounces, but you can get lower fat types, for a higher price. Beef and pork are generally higher calorie but you can make a difference in how you cook it. Fish is probably lower calorie, but isn't in my budget (and I'm not a big fish fan) so I haven't tracked it. I can get a quart of plain yogurt for less than $2, and I use it in place of sour cream. It takes a little getting used to, but it's really good on a baked potato, lower calorie and fat, and cheaper than sour cream. It's also helpful with digestion if you have problems like I do. Fat free milk is a good protein source, but make sure you drink it with a meal with some fat in it to get the most benefit from the nutrition.