It may never work perfect, as you're using a non-sis freewheel. Also, it sounds like you may be using a freewheel whose biggest cog your derailleur can barely handle, if at all. The 5-speed indexed shifters are usually pretty junky and the benefit small over 5 speeds. And mixing used/new freewheel/chain is generally a bad idea. But you can try these steps to make it work.

1. Detach the cable from the derailleur.

2. Adjusts L limit screw so that the upper pulley lines up with the small cog. Look at the derailleur from up and behind. if the pulleys and cog do not line up, your hanger is bent. You can try to bend it back but DO NOT GRAB IT BY THE PULLEY CAGE (this is assuming you're using a derailleur with a claw). Hold it by the upper assembly, or better yet try to find something to grip the claw itself. Align and readjust limit screw as necessary.

3. Put shifter in hardest gear, screw adjusting barrel on derailleur down most of the way and reattach cable. Shift 1 speed while pedaling. if it doesn't shift up, back off barrel adjust until it does. Work up to big cog doing the same thing. Adjust H screw so that it doesn't run off the big cog.

If the upper pulley still grazes the big cog, taking a link out of the chain may make it work.