It means that you REALLY REALLY REALLY want a cup of tea. And once you drink it, there will be no evidence of its existence anymore, so you murdered it... right? (I could be wrong :-D I am just messing with the expression, my dear tea murderer! ;) ) --------- and coffee?? yes... just shoot it in the head ^^

It's an idiom which means "to really want something", usually food or drink (due to greater than usual hunger or thirst).

Correction: "I'm so thirsty I could murder FOR a cup of tea/coffee etc."

I believe murder in this case means to gulp the tea down fast, like you would gulp water down. Usually, you sip tea, slowly.

The person wants tea (or coffee) so much that his/her feelings are like that of a crazed murderer, figuratively of course. He/she is desperate and really, really wanting it. Not many murders are the result of someone being denied their cup of tea :) No cups are harmed in the action.