Go back to the time when Ulfric was captured by the Thalmor and interrogated. When the Thalmor found out he was the next Jarl of Windhelm, they tried to use him as a spy. Some people believe while he was captured he gave information that allowed the Thalmor to capture the White Gold Tower. This is possible considering the fact if you were being tortured you do anything to save your ass. But in my opinion I do not think this was true at all since, Ulfric was not a Jarl at the time so his information reguarding the White Gold Tower would be rather limited. Secondly he's an easy escape goat since some people already do not like him for going against the Empire.
Except that info that he gave was unimportant in the grand scheme since the Dominion had already taken the Imperial City. All they needed was to break him which they did. Also are you talking about in-game because it seems that no one knows that he gave the Thalmor any info.
With this all said, the Thalmor were hoping to use him as a spy, but that did not work out as plan. Ulfric took back Markarth from the Forsworn. This would be an issue for the Thalmor because that was one less city to worry about worshiping Talos and to "take" over. I would assumed this pissed the A.D.off-very much.
They weren't looking for a spy but an agent to destabilize Skyrim. The Thalmor wanted the Markarth Incident to happen since this allowed the Dominion to send in the Thalmor to pursue Talos worshippers and to add even more of a reason for the Stormcloaks to fight against the Empire .
Now Ulfric goes a step further and challenges the High King to combat and defeats him. Again this would piss the Thalmor off. The high king took orders from the Empire and the Empire took orders from the Thalmor. Now Skyrim is in a state of chaos. The Empire needs to gain Skyrim back because the Thalmor need to remain a tight noose around the Nords, especially those who worship Talos.
Ulfric killing the High King wouldn't upest the Thalmor since they needed the civil war to weaken the Empire.
I presume the real reason the Thalmor want Ulfric dead is because he knows too much about the Thalmor and he refused to be a spy for them. The Thalmor cannot simply come out and give the true reasons for wanting Ulfric's head cut off, so they can use the excuse, he worships Talos and defies the Empire. But really when you go against the Empire (now a days) you are going against the Thalmor too.
This statement is just wrong the Thalmor don't want him dead. They want Ulfric to weaken the Empire with this civil war and they even tried to intervene in order to stop the Empire from killing Ulfric.