Andrew R Stewart 

One needs to separate the various reasons a gear system can "skip".

Start with the chain. Does it have tight links? Relaxing the rear der cage spring tension while back pedaling will let a tight link be noticed as it travels through the cog/cage path. Does the chain have any twisted links? Are any side plates pulling off their pins? And the obvious possibility of 'stretched'/worn chain.

Next are the cog/ring teeth. Any twisted/bent? Are the cassette/freewheel cogs all flat against each other with proper spacing between each? Are any cogs broken/cracked or missing teeth?

Next is the ders (rear in this case). Is the cage/pulleys hanging vertically? Are the pulleys straight and intact, no distortions or cracks? Does the chain have clearance through the cage in all cog/ring combos? Are the limit screws set properly (doubt this is the case given the description)? Does the rear der match the cog count and the lever indexing count?

Then the cable. is it routed through the housing and end caps well? Is there undo friction in the path (a much more common problem then many would think)? Is the cable anchored on the correct side of the anchor bolt (another common issue)? Is the cable of the right diameter (1.2/1.1mm)? Is the cable housing the correct compressionless type? Is the cable beginning to fray, say just after it's head in the lever and out of sight? And, of course, is the cable tension set properly?

So there are 18 aspects of index systems that need to be right. Almost every problem is found within this list. Now you need to start looking for your problem. Andy.