You can eat just as cheaply on a diet as you can when you don't diet
The stigma that healthy food is expensive comes from the fact that things like fresh fruits and vegetables in large quantities can run up your food tab. And protein sources like lean meats are relatively expensive. And most notably, pre-packaged 'healthy' food is incredibly overpriced. But the same people who will tell you that they can't eat healthy because its too expensive, I bet them same people won't bat an eyelash at dropping 7 bucks on a McDonalds value meal. I can eat an entire days worth of healthy food even when I am bulking and purposefully overeating for under 7 bucks. When dieting, I can easily keep it around 5-6 bucks if I want to.
There are a few rules you need to set for yourself to get the most for your money. Start out with figuring out your protein sources. What is the cheapest meat for you? Can you find chicken breasts for under $3/lb? If so, that is probably your best bet. Aim for 8 oz (pre-cooked weight) of the cheapest lean meat you have available to you per day. You can split this up into 2 meals if you want, but your portion per meal will be smaller obviously.
Now that you have gotten your main protein source out of the way, you need to think of supplementary sources. For the average inactive person, the extra protein they randomly pick up throughout the day from dairy, beans, nuts, etc will probably be good enough. But if you are starting a weight lifting plan, you will probably have greater protein needs. This can start to run up your grocery bill a little higher. Protein powders run anywhere from 75 cents to a dollar for a serving unless you go with a bargain brand like that Wal-Mart shyt which doesn't pass the quality test. Eggs are a great value when on sale for less than $1.50. Milk is cheap, but you have to watch your calories closely when you are drinking calorie sources and it is not recommended if you don't have a well thought out plan.
If you are looking to eat a lot of protein, your grocery bill is going to creep closer and closer to 10 bucks per day if you aren't careful.
Now, the next question is what else to fill your diet out with. My rule of thumb is "If it is not a major protein source, it must cost less than $1 per 600 calorie serving size." You can choose to adopt this rule or not, but it has worked out well for me to stay on budget. It is not always able to be followed though.
Cheap sources of great dieting carbs such as Old-Fashioned Oats and Long Grain Brown Rice are well under that threshold. Whole potatoes start to creep closer to that price, but I hate cooking potatoes anyway so I don't buy them. Fresh fruits and vegetables probably will not fit that price range, but I will usually make an exception for a serving or two of vegetables. And I always make sure I'm taking multi-vitamins to cover any deficiencies that may exist.
Cheap sources of great dieting fats like peanut butter and olive oil fall well within that price threshold. Nuts are usually more expensive other than the super cheaps ones like cashew halves. You won't be eating a ton of fats unless you are on a low-carb diet, so this doesn't matter as much.
Too Long; Didn't Read Version: Expect to spend anywhere from 5-10 bucks per day. That is a wide range of prices and could potentially double your grocery bill. Make good choices, always watch store ads and look for good sales on healthy items. If you are spending more than $10 per day you are not shopping smartly...period. And $10 per day is a small price to pay for health and well being, especially considering that fat people will spend $4 on a soda and small bag of chips then bytch about being hungry again 30 minutes later and then complain that eating healthy is too expensive