Couple different things, and a lot of it depends on your ability to be tactful and any given specific relationship with a client.
Keep a file with all of the notes as a CYA measure in case you ever need it.
Best time to bring up the notes sent to you but not your supervisors is during a review, even if just in the aggregate while filling out the self-evaluation part ("This last quarter/year/whatever, I received numerous of letters thanking/praising/acknowledging the work I've done for clients."). If your boss questions that, you have the CYA file full of the notes you can share.
As far the step of having others send notes to your manager, best way is to also tie it to your review if you have that kind of relationship with the client. "Hey Jim, thanks for the note. Always glad to help and hear that I'm providing the service you need. I've got my review coming up soon and was hoping you could do me a solid and pass that note onto my supervisor so that they have a better understanding of the quality of work I'm providing on a daily basis. If not, no worries. Thanks" But again, only do that if you have the type of relationship with the client/customer that it'll work and if you can phrase it tactfully.