As a person who is doing a '64 Vette (mine is a 'vert), here is a few things to consider.
1. If the birdcage is rusty - run (do not walk) away. I'm stripping mine down to have the birdcage acid dipped and I'll be TIG welding all of the bad spots then having it powder coated the correct colour. It's a lot of work and I have purchased the vast majority of all new panels for this type of job, but if you are not into the effort then don't do the project (my timeline is measured in years).
2. Consider what you want. If you want something that can hang corners, don't get a numbers matching car. It's a bit of a waist. I'm putting a SRIII frame under mine and I have a near perfect frame / restored engine / trans / rear end that is up for sale. None of the numbers matched so I'm not that worried.
A SRIII frame with 2010 ZR1 suspension will be sweet...... And a few minor mods
3. If you are going down the numbers matching route, check the frame (especially in the back pockets where the rear arms attach). Lots of rust or worse.
4. From under the car looking up at about where the package tray is behind the seats, look for a crack. Mine is cracked there and I'll have to mend it when the birdcage is out. From what I've been told, it's common for mid year bodies to crack here.
5. As folks have already said, '64 is the least desireable C2. It is not a split window '63. '65 have disc brakes and started the big block era with the L78 396 option. '67 is considered the ultimate mid-year.
That's not to say they aren't cool. Mid-years are cool (IMHO).
Something to consider is putting a shout out to the folks in the C1 / C2 area of
www.corvetteforum.com. If there is someone in your area they may be willing to drop by and have a look.
Be aware that with the Corvette name comes a price escalator for parts that may cost less on a different vehicle. It's just part of owning a 'vette.
Thoughts?