Cheese suggestions to replace melted Velveeta

To maintain the consistency of Velveeta you would be well served to start with a base of Cream Cheese into which you can melt other soft cheeses. You have a wide range of choices. Cream cheese will promote the right texture and is a neutral enough flavor that what ever you want to add as a 'feature' cheese(s) will come through.

Personally, I might look at a smokey Swiss & Cheddar I get at my local deli or for a little kick some pepperjack.

If you want the texture of Velveeta, you have to make your own processed cheese. The best way I know is to mix finely grated cheese with some kind of milk or cream, gently melt, and add gelatine to make the sheets. You can find the story of perfecting the process, as well as the final recipe, on Kenji Lopez Alt's column on Serious Eats, http://aht.seriouseats.com/2011/09/the-burger-lab-how-to-make-super-melty-cheese-slices-like-american.html.

If you wonder "but why make processed cheese if I want to get away from Velveeta" - if you use some good quality, aged aromatic cheese as the source, the end product tastes differently from the stuff you can buy.

You can probably modify any cheese to melt similarly to velveeta.

Sodium citrate would be my go-to method. It's actually one of the main things that gives Velveeta its smooth, processed texture, but you can buy it yourself and add it to a cheese of your choice. (A good alternative would be gelatin plus (evaporated) milk; see rumtcho's answer for a description of that. It's not quite as close to processed cheese texture, but it should be good enough for your purposes.)

The basic process is to heat a bit of liquid (water, milk, broth, beer, whatever you like) with sodium citrate in a pan to dissolve it, then slowly blend in grated cheese, letting it melt, until it's all incorporated. Ideally, you use an immersion blender to get it really smooth, but I think determined whisking would suffice. For thinner sauces, you can use more liquid. I've also seen varying ratios of cheese to sodium citrate, anywhere from 25:1 to 90:1.

Here's an example generic recipe for just the cheese. I've seen this kind of thing used generally for sauces, for example this Serious Eats "Modern" Mac and Cheese recipe or this nacho cheese recipe, but it should work for you too as a melted processed cheese replacement.

That example recipe says to use immediately; it'll be like melted processed cheese at that point. If you let it cool, I'm pretty sure it'd solidify into something like Velveeta.

Note that if you want it to be really thick, you could likely use slightly less liquid, as long as it's enough to cover the bottom of your pot and dissolve the sodium citrate. You could probably use a little less sodium citrate too and still have it be plenty smooth while seeming a bit less processed, but since you're actually trying to replicate processed cheese, I guess you might as well go all in!

The ideal answer here is to use Sodium Citrate to make a cheese of your choice soluble in water. This Modernist Cuisine article is exactly what you want.

If you look at the ingredients in Velveeta or most American cheese slices you'll see that they use Sodium Citrate to emulsify the cheese and water. Most of the other techniques here, including your bechamel concept, will dull the flavor of the cheese, and in many cases fail to approximate the texture of Velveeta. It sounds like the Serious Eats approach mentioned in another answer would achieve something similar, but it's unclear to me why the author explicitly eschews the use of Sodium Citrate.

In any case, the technique I'm suggesting will allow you to get the exact consistency and stability of Velveeta with the minimum possible effort and without masking the flavor of your base cheese at all. You can get Sodium Citrate from multiple suppliers on Amazon or elsewhere online.