LATECH right on the money. Full-frame cars of the era are FAR superior to build into good street cars. The GM A-bodies are obviously the most plentiful, and probably the best overall basis. If you really want a 4-door sedan (avoid 4-door "hard tops" for performance), it should be pretty easy to find at a reasonable cost, as they're the least "popular" of the styles. The Ford and Chrysler offerings were all "uni-body" except the BIG cars.
Understand that ALL the performance "goodies" for the suspension that fit the 2-door coupes and hard tops also fit the 4-door. Don't be fooled. While these cars LOOK big and heavy, they really aren't. They average around 3,500 lbs. unless "fully equipped". Linear traction is among the best ever for a production "family" car ("4-link", coil spring). Lateral traction is VASTLY improved with modern suspension components readily avialable. Add a "quick ratio" steering box and you have a FUN car that really does "handle". Talk about a sleeper...
Engine options are wide and "varied" as well. The small block Chevy offers peppy performance at a reasonable cost. Both the big block Chevy and Pontiac can provide SERIOUS torque and HP for an "affordable" price. The big Olds and Buick engines have some advantages, but are generally more expensive to build reliably. Less availability for higher level performance parts, as well. If low-end torque and light weight are an appealing combination, the Buick 455 is the better one. It IS limited in ultimate power output and RPM due to the lighter weight. ALL of these engines literally "bolt right in".
The "plain jane" Chevelles, Cutlass, Tempest and Skylark are the ones to look for. They usually have smaller engines (don't be shy about 6-cylinder cars), less options and were true "family" cars (less abuse than the muscle cars). EVERYTHING that makes a muscle car a "muscle car" fits these cars.
Just a thought or two...