Do you really want the cheapest stuff you can get or do you want something that will taste good?
If you want a really good cider, you have to start with good juice. If cheap alcohol is the overiding concern, your best bet is to dilute a big jug of corn syrup with some tap water and ferment that. Although once you factor in the value of your time, youre probably better off just buying cheap wine or malt liqueur.
The types of apples used to make sweet apple juice, including concentrate, most store bought juice and even a lot of the roadside fresh stuff - are high sugar, low acid, low tannin varieties, such as Red and Yellow Delicious. They are the apple equivalent of a table grape. They taste good fresh, but when you ferment off the sugar, there isnt much taste left. You wont see anyone from a self respecting winery shopping for their grape juice at Costco or Walmart why should you?
Some folks have posted good results with generic juice by adding tannin,, acid, malt and other stuff. Im sure that is possible, although I dont see the point, unless you have no way to get good juice.
If you want something that will taste good without additives, you need to start with a little research to find out what varieties are grown in your area that will give you some acid and tannin. For starters, check out:
The Best Apples to Buy and Grow
WSU Mount Vernon Northwestern Washington Research & Extension Center
Then call around to the local presses to see what they will sell you. Tell them you want to make cider. The website all about apples has pretty good listings for orchards in each state, as well as info about a lot of apple varieties.
All About Apples | Apple Orchard and Farm listings
Most presses even the commercial ones - use some sort of 'flavor' apples to give some zip to whatever cheap bulk apples they are using. Those are usually the ones you want. In Central Virginia, these include Staymans, Winesaps, Jonathans, Yorks, Granny Smiths, etc. Some presses can get apples that are specifically cultivated for hard cider. Northern Spys, Pippens, Russets, Black Twig, etc. Those are even better but usually more expensive so its a bit of a trade off, depending on how much you are making
Like wine grapes, the best cider apples are very much a matter of personal preference, so its probably worth checking out a few different presses to see what they have. If you want total control, you can press your own, although thats pretty time consuming.
Around these parts, the local presses sell bulk juice made from good cider apples for under 3 bucks a gallon, although you generally have to get 30-50 gallons to get the bulk rate. Single gallons are more like 5 to 7 bucks.
The yeast is just as important as the juice many will strip the flavor out no matter what you start with. Ive had best results with Ale yeasts.
If fresh pressed isnt an option and you go with store bought juice, you might want to check out what is available and see how it tastes fresh, before fermenting anything. I tried fermenting some Orchard Valley juice, which Whole Foods carries. It was drinkable, but not nearly as good as the fresh pressed stuff. You also might want to consider stopping it earlier, say between 1.010 and 1.020, because once the sugar is fermented out, there isnt much else left.
In Virginia, unpasteurized juice can only be sold at the press. If you go with pasteurized, make sure it is UV and not heat treated. UV sets the pectin a bit, so the cider wont clear as well, but it doesnt affect the taste. Heat treating knocks a lot of the flavor out.