A House of Cards (Solution)
It isn't necessary for each team member to play a role, but the puzzle is easier to solve (and more fun) if all four team members participate. Spade starts the conversation, and if everyone speaks exactly when they're supposed to (according to their scripts) it goes like this: Whoever played Diamond should have written down all mention of card suits. Every mention of a suit is bolded above, and they are named in this order:
Whoever played Heart should have kept track of how many times each character was accused. Heart was accused 4 times, Spade 2 times, Diamond and Club one time each. There are eight suits mentioned, and there are also eight characters at the end of Greenstreet's letter, ETDBXKPR. If you replace each suit with the number of times they were accused, and subtract that number from the character in Greenstreet's message, you get the answer:
SSCDHSCH => 22114214