How do I tune slipping gears? My kid's bike has some gears that aren't working properly (rear derailleur). They "slip". That is, he's pedaling along and suddenly the gear slips, as if it is going to change gears, but doesn't actually change gears. This happens repeatedly. It doesn't just slip once and then stay, no, it continues to do it over and over again while pedaling without making any adjustments on the gear selector.

Another interesting symptom is if I try to keep track of which gear it is supposed to be in, and which gear it is actually in, I get a table something like like this:

Gear it's supposed to be in - Gear it's actually in
1 - 1
2 - 1/2
3 - 2/3
4 - 3/4
5 - 4/5
6 - 5


... where the "/" indicates that the gear state could be one or the other, sometimes depending on whetheur it was shifted into from the higher gear, vs shifted into from the lower gear (shifted up or down) and/or the two gears it is slipping between.

How do I go about tuning them?
First got though  this park Tools guide

If you are still having problems, I suspect the derailleur cable. Either replace it (best option as they are cheap) but its worth flicking the outers off the bosses, cleaning and lube as much of the cable as possible.  If this improves things, a new cable may be needed. 

If this does not help you probably need to check the derailleur alignment. unfortunately your derailleur has an integrated hanger so its harder to check. Fortunately, as its only 5 speed, accuracy is not so important. If noticeably bent, you can straighten it using an adjustable spanner, but will have trouble getting it accurate. 

Another possibility is the derailleur is flogged out, although in my experience kids don't change gears enough for this to happen