There are no independent statistics to show which are the best PPI claims handling companies. Of course you can always ask around your friends if they have made a claim and if so what their experiences were. By and large the most successful PPI claims companies have a success rate of over 90%
However, its important to remember that the best company may not necessarily be the one that claims the best success rate. If after they have reviewed the details, a claims company could feel that your claim is only borderline, they may refuse to take your case on. This of course means that their success rate is kept high by only taking on the cases which they think have a very high chance of success. So if your case is borderline it may be that a company with a lower success rate actually will take on your case and may be successful. Bear in mind that of all claims referred to the ombudsman, whether made by claims handlers or individuals acting for themselves, around 70% are successful
Of course best can also be measured by the fees charged. Is a company with a 95% success rate but that charges 25% better than one that has a 90% rate but only charges 15%? Finally best can also be measured by how good the actual service is that you receive, for instance if you are given a dedicated claims handler for your case you may feel that this is a better service than a company where you speak to someone different every time.