This isn't a derailer problem. You have all the symptoms associated with replacing a very worn out chain: Basically, your freewheel is worn out (the gears in the back). Typically the smaller cogs wear out faster, because they're used more and because the smaller cogs have fewer teeth to distribute the wear. I'm guessing that the worn cog is the one that you use most
The old chain wore out, and the old cog wore out to match the old chain, so it didn't skip. But the new UNWORN chain has a slightly different pitch from the worn cog, and it therefore skips. Read the "Chain and Sprocket Wear" section of this article:
http://sheldonbrown.com/chains.html
You should replace the whole freewheel. If you don't, the new chain will wear out very quickly. In general your shifting will get worse as the drivetrain continues to wear. The best thing to do is to get a new freewheel. Since you're using friction shifters, you may be able to upgrade to a 7 speed freewheel in the process. That will give you a few extra gears. In any case, you can get modern freewheels (which shift much better anyway) pretty cheap from nashbar.com
You can prevent your drivetrain from wearing out too fast by replacing the chain whenever it elongates by 0.5%. The way to tell is to measure 12 links, and if they're longer than 12-1/16", then replace the chain.
I hope that helps...