Although I have not removed an automatic transmission from a 90 Vette, I was able to successfully remove, rebuild and replace the 4+3 manual from my 85. I was able to do this in my garage using jack stands but it was not the easiest repair I have ever performed.
Some tips you might find helpful:
1 - Get some big heavy duty jack stands. You are going to spend a lot of time under the car and you want to be safe.
2 - The car will have to be raised high enough to slide out the transmission, which means that you will have to raise the car in stages. If you simply raise one side of the car up really high and place it on stands, there is a chance the car will fall off the stands as you start raising the other side. Be patient and careful here.
3 - The driveshaft will have to come out. Not a big deal, mine came out easily even though it had not been removed for 20 years.
4 - The torque arm will have to be disconnected and removed. The bolts on mine could not be removed with hand tools, I had to use an impact gun. If you don't have access to an impact gun, you may have a problem here. Not saying you can't get the bolts out, I'm just saying that on my car they would not come out without an impact gun. You will also need a box wrench to hold the nut on the top of the torque arm as you work to loosen the bolt from the bottom. (Note: the box wrench really hurts when you drop it on your face, I know from personal experience).
5 - In order to remove the bolts at the top of the bellhousing, I had to remove the following: both valve covers, plenum, runners, distributor, oil pressure sending unit, and fuel rail. Removing these components gave me access to the bolts at the top of the bellhousing. You may be able to get by with removing fewer components, but I found I needed to remove them all.
6 - I used a regular floor jack to support my transmission as I removed it, but I needed one person to handle the jack, and 2 people under the car to get the transmission out. As I said, mine was a manual, and I believe that the manual is lighter than an automatic. I found my 4+3 to be quite heavy, so be careful if you decide to remove the automatic. I found that getting all of the bolts removed was the biggest challenge, and the actual removal was not that bad since I had help.
If you want to get your car fixed and running again quickly, I would recommend getting a shop to remove and replace your transmission. I did mine in the garage because I was determined to rebuild the transmission myself, and my car was out of unavailable for almost 8 weeks.
Note: When I was done, I promised myself "never again". But if my transmission were to fail tomorrow, I would probably do the work myself again.
something to check into would be Beam Plates from zfdoc.com they sort of take the place of the sealant ... help firmly secure the C-beam. Two plates with the nuts secured so getting the proper torque is a lot easier. Supposed to help keep the rear from kicking to the right under hard acceleration ... more noticeable with improper torque on the Beam bolts. Plus you don't have to worry about eating a 18 or 21 mm box end .... trust me it will happen. 
