I'm no engineer, but as an engineering enthusiast hopefully I can help you steer your way towards a car that will fit you well. Choosing carefully now will help you avoid the automotive purgatory of a stalled restoration and the depression that follows!
Step 1: Decide now what you want the finished car to be. Before you buy one single part for the car decide what you want in the end. A high dollar numbers matching restoration? A highly modified corner carver built with the latest technology to upgrade older Mopars to hang with the modern cars? Or a drag car that can eat up a 1/4 mile in no time. You can ultimately blend elements of the three but to start you should know what road you're going down.
a: Restoration. Find a numbers matching car and restore it back the way it came from the factory. If it's a green 440-6 car with 4 speed then so be it. You are now a curator of this exibit and should maintain it as such. But in return, the value of your car will maintain or rise as well.
b: Modification for handling: This too is a very expensive option since it involves much aftermarket parts and fabricating. This method can be done in stages but sometimes the whole front and rear suspension and portions of the frame could be replaced in this scenario. Some people repower them with modern Hemi engines and drivetrains. Even a custom interior is not off limits. It blends the best of the old with the cutting edge of the new and if built properly will give you a similar road experience to a modern car. The draw back is that some people at shows may not share your vision or enthusiasm for your project.
c: Modification for racing: This is one of the oldest types of modified cars you'll find. Roll cage, gutted out and tubbed. A lot of the time they're found without engines (for obvious reasons) which allows you to repower them however you want. Supercharged Hemi? Sure! The goal is going fast and the car is almost secondary.
OK now having narrowed down your main goal, you may want to consider mixing and matching what you want. Now on to car selection. This depends on how much work you can do yourself, or how many friends with shops owe you favors. Metal work can get really expensive really quick. You can buy an Challenger bare shell or replace quarters, trunks and floors that are rusted but it all takes time, patience and money. The less time and patience you have, the more money it costs. That said, I recommend finding the least rusty starting point you can. If you're building a modified car and are not concerned about originality you might consider looking into a 72-4 Challenger and having the grille, marker lights and tail lights switched if it saves you money. I do NOT recommend picking up a numbers matching big money car and modifying it. In the end you'll have lowered the value of the car and maybe upset someone who would have enjoyed the car as an original.
Time and cost: This is hard to figure since it depends on what you started with. Picking up a rust free 73 Challenger and converting it to what you want may cost less than an original 70 rust bucket. An all out resto of a big money car can easily end up double or triple what it's worth when done. And of course it takes far longer to come up with the special NOS parts that a true restoration requires.
In the end it's your car. 20k will get you a good start down whatever road you decide. Just choose wisely! It will help you further on down the road.