A letter can be good but don't make it too long or presumptious. Just a basic "I'm training for a CPL and would love the chance to visit the flight deck" would be fine.
Don't need your licence or logbook really.
The advantage of a letter is that the correct info comes to us. Sometimes you might tell the cabin crew that you are a pilot training for a CPL but we don't get that message. And if we get a message that "someone has requested the jumpseat for take off" that doesn't go down too well.
Sometimes there are reasons we can't get you up the front. For take-off its usually because we got the message too late or you asked too late. During the flight you might have asked too late. Or there might be some training going on. Although usually if you get turned down for a visit during the cruise I would say its because the Captain can't be bothered.
I have asked when passengering on other airlines aircraft and got the response "sorry they are training today". Only 2 pilots are announced in the PA so its either a new Captain or a new FO. There is no real reason not to have someone come for a visit during the cruise.
Often we will have a visitor up during the cruise and, if we know they are keen and in training for a commercial, we may invite them to stay for the landing. But we like to meet you first, so its difficult for take-off. The general rule on the flight deck is don't talk below 10,000 feet and try and listen for radio calls. Short answers help.
If you can get a message to the company before you fly that helps too. You can send a letter to the airline before your flight or if you know people in the company ask if they would put a message in the operating pilots pigeonholes - usually let both FO and CPT know because if there are any roster changes one or other might not get the message.