The first thing to know is that it takes a lot of time and due diligence. Before he picks a horse out to claim he attends every race, every day at the target track. He believes it is folly to try to pick a horse based on his race record alone.
So the basic steps after committing to watching every race in person are:
1. Handicap the race to determine which horses in the race are worth the claiming price.
2. Watch each horse carefully in the paddock, check off, in the program, the horses that are sound and look the part of an athlete. Make notes next to each horse in the program about flaws such as tendons and joints that have current or past problems, conformation faults, foot problems, and obvious lameness. He will also note if a horse looks dull, has skin problems, underweight or overweight, bad trimming or shoeing, because these things can definately be improved and may allow him to move a horse up.
3. Watch the horse jog off in the post parade to spot bad movers, bad actors, lameness, etc. and make notes in the program again.
Invariably by the middle to end of a race meet, he has chosen several targets to claim and has put money in the office to cover the projected claiming prices.
When a claim target comes up in a race with the projected claiming price, hopefully dropping down from a previous level, he does not drop a claim until after he has watched the horse walk from barn to saddling paddock, or if that is not possible while it is being saddled and has walked off, to make sure the horse is sound.
The experience to spot lameness, old injuries and conformation faults is crucial to a successful claiming program. I'm sure that many trainers and owners claim based solely on past performances, pedigree and "numbers". This method is NOT recommended for the average small scale trainer and owner.