"Lok'tar ogar! Victory or death - it is these words that bind me to the Horde. For they are the most sacred and fundamental of truths to any warrior of the Horde.
I give my flesh and blood freely to the Warchief. I am the instrument of my Warchief's desire. I am a weapon of my Warchief's command.
From this moment until the end of days I live and die - For the Horde!"
This is the Oath. This is what has been part of the Horde since it's clans days. The Orcs back on Draenor followed a similar Oath as Durotan could not tell Doomhammer, his best friend, about the Corrupted Blood of Mannoroth because Doomhammer would be obligated to tell his own Chieftain.
Now: Did Garrosh break the Oath first? Nope.
Vol'jin threatened Garrosh right from the get-go. An unwarranted threat on his life, and despite this, Garrosh let Vol'jin live when in all reality he could've executed him. Who gets away with threatening the leader of some super-power and then is left in power? Garrosh showed lenience here.
Through Cataclysm, the Darkspear did nothing for the Horde, because of Vol'jin and Hellscreams exchange of words. So to Garrosh, the Darkspear are pretty useless. They've played little role in Cataclysm, except to interfere with some obscure Troll plot, and have offered nothing but hostility to Garrosh.
Theramore comes along and Garrosh summons the entire Horde to march with him. Vol'jin is forced to go but still continues to go against Garrosh. He organizes several 'secret' meetings with others leaders, questioning Garrosh's tactics. Now, is this as bad as betrayal? No. But the proper thing to do would have been to bring it to his Council, as Eitrigg and Cairne did. But instead he met in secret and offered nothing to support his crumbling relations with the Warchief.
And finally, after showing up -uninvited- to Pandaria, Vol'jin disobeys Garrosh's orders in the Saurok cave, refusing to gather intelligence on the Mogu-Flesh shaping magic that Garrosh wanted to learn more about.
Vol'jin had everything he got coming to him. Openly threatening the Warchief, not supporting his Campaign, speaking out in his attempt to rid their front doorstep of enemies and now disobeying his one order that was meant to 'test' Vol'Jins loyalty to the Horde. I'm sorry, but as much as I'm fond of Vol'jin (since we've been with him since the RTS days), he broke the Oath first.
The oath is dedicated to the Warchief and the Warchief alone. When Orgrim Doomhammer killed Blackhand, the Horde shifted their loyalty to Doomhammer because he was the New Warchief.
When Thrall became Warchief, he enforced the Blood Oath (not as violently), but the Oath was there. What happened when the Forsaken turned on the Horde? Thrall struck back. He marched into Undercity and placed his Kor'Kron there because he did NOT trust the Forsaken.
Suddenly Hellscream tries to enforce the Blood Oath and he's seen as a Bad-Guy? Because Vol'jin is a 'Hero' and best-buddies with Thrall, his actions against Garrosh can just be ignored because 'he's not a meanie like Garrosh'? I don't think so. In regards to the Blood Oath, Vol'jin broke it SEVERAL times and the fact that he was assassinated (or attempted on) so late in the game is a surprise. I expected Vol'jin to go long before Pandaria.
To everyone saying the Blood Oath is 'just words' and that Vol'Jin's threat is 'just words', they're not. The Oath is an important part of *Orc* Culture and was adopted into the Horde. Which means, Orc or not, every Horde member must swear it, which means they are swearing themselves to the Warchief.
That is why Garrosh sees himself as the Warchief of the 'True Horde'. His loyalists and supporters are keeping true to Orcish custom.
Does that mean they're all insane? No. Look at Nazgrim. Nazgrim's loyalty to Garrosh is tied strictly to the Oath and to his duty. He has no love for Hellscream on a personal level. It's the chain of command. Garrosh is his Warchief and by law and blood, he is sworn to obey and carry out his will. Breaking the Oath would be difficult for an Orc. It's breaking the one rule that you have followed your whole life, the one rule you've bound yourself to.
Garrosh's assassination attempts on Vol'jin are completely justified (even though I'm glad he actually survived).
And don't get me started on that Hypocritical piece of Green Garbage known as Thrall. He's more of a traitor than Vol'Jin is.