Damn
You will get killing spree when you have 3 kills in a row WITHOUT dying. 4 kills will be a dominating streak, 5 will be mega kill and so on. Here's a little help for you:
Killing Spree - 3 kills without dying (that means in a row, you kill one now, another after 2 min and last one after 5 min, you didn't die in that period)
Dominating - 4 kills without dying
Mega Kill - 5 kills without dying
Unstoppable - 6 kills without dying
Wicked Sick - 7 kills without dying
Monster Kill - 8 kills without dying
Godlike - 9 kills without dying
Beyond Godlike - 10 or more kills without dying
When you hear a guy screaming :"The Scourge or the Sentinel are OWNING" that means your team killed 5 or more enemy heroes in a row without dying (all of your team members was alive in that time when you or anyone from your team killed 5 enemy heroes, example: Sentinel kills 6, Scourge kills 0)
When your hero dies, you'll respawn in some time and then you'll have to start the whole thing over again, back from the 'Killing Spree' (you first need to kill 2 heroes to gain killing spree after 3th).
First Blood - Getting the first kill of the game.
Double Kill - Getting two consecutive kills by the same player. (if you kill 2 enemy heroes with same spell or in 10 seconds interval for example)
Thriple Kill - Getting three consecutive kills by the same player.
Ultra Kill - Getting four consecutive kills by the same player.
Rampage - Getting five consecutive kills by the same player.