First off, you need to replace that rim and the spokes. The spoke nipples from that period of rims only go through the first wall of the rims, and once they have significant warp, they are toast. You can spend lots of time truing it, rebuilding it with new spokes, etc. It will be well worth your time and money to get a new rim built around the existing or a new hub.
As for re-building a new back wheel, I may have some bad news for you. Many 6-speed aluminum-framed or lugged bikes from the same period can't be bent out to accept the more modern 130 mm wide hub-sets. The axle drop-outs for the 1988-1990 6-7 speed rear cog hubs are 126 mm wide. They just stopped making those very soon after that (in favor of 7-9 speed cassette hubs), and it is very, very, very difficult to get a new back wheel with a new hub. The only company that makes new ones is in that size is Phil Wood, and they are damn pricey (about $130 for a naked hub with no cogs, spokes, rim, etc.).
I think that your best option is to get a more modern rim and build a wheel around your existing hub with new spokes. This will cost you in the $120 and up range for parts and labor around here, depending on how tough or light you want the rim to be. If you want to try it for yourself, newer and super-tough rims are about $30, and I don't know the price of the spokes, but I'm sure that they are no more than a dollar or two each at most (probably 32 or 36 spokes on that tire).
If you don't like the bike that much, I would also consider just looking on CL or at local garrage sales for another bike that you DO like. You will be able to find some old Japanese chromolly bike that works just fine, for cheaper than we are talking about above. You may want to cut your losses.
Later.