In the last episode of Supernatural Season 10, Dean summons Death to kill him, so that he will not be turned into Demon due to Mark. However, Death tells him that he cannot kill Dean with Mark as it will free the 'Darkness'. However, he can send Dean far away and Dean needs to kill Sam, so that Sam will not try to bring him back.
However, he changes his mind at last moment and kills Death instead of Sam. But why kill Death, he can simply leave Sam.
At this point Dean has essentially 3 choices:
Do as Death asks, kill Sam and go "away"
Take Death up on his first offer - to give the Mark to someone else and then have Death remove it from himself.
Kill Death and figure it all out later.
Number 1 is out since he can't bring himself to kill Sam (and he knows that if he doesn't do it Death will in order to prevent Sam trying to "rescue" Dean), Number 2 is out since he knows what the battle to control it is like and doesn't want to put that on someone else who will likely kill innocents and quite probably get himself and or Sam killed trying to stop them. He can't just call the whole thing off since Death will still kill Sam to prevent him attempting to remove the mark (and release the darkness)
Even if I remove Dean from the playing field, we're still left with you. Loyal, dogged Sam, who I suspect will never rest until he sets his brother free, will never rest until his brother is free of the Mark, which simply cannot happen lest the Darkness be set free.
That leaves Door number 3 (the "figure it all out later" part becomes sort-of moot since unbeknownst to Dean Rowena/Crowley/Castiel are in the process of removing the Mark) but it's in keeping with Dean's character which is that killing things is his default plan, and of course his devotion to Sam above all else.
Dean killing Death was an accident, which is why he looks so shocked. He meant to decapitate his brother, but missed and accidentally hit Death who unwisely happened to be standing in the path of the scythe. The scythe looks pretty heavy and Dean's probably never handled one before, so it makes sense that his swing would be off. It looks like his depth perception might have been inaccurate, which also makes sense since he was very emotional at the time and wouldn't have thought to check again.
