Why did Daenerys Targaryen feel betrayed by Doreah?

Let's see the situation from Daenerys's eyes:

1. She returns to the palace to find many of her men and a faithful handmaiden (Irri) dead.
2. Not only that, but her precious dragons, her babies, missing. Her other trusted handmaiden (Doreah) is also missing.
3. She spends many days, nervous and afraid, as she looks for the missing dragons.
4. Once she finally finds out where they are hidden she has to go through an extremely dangerous and stressful ordeal (the House of the Undying) in order to get them back.
5. When she figures out that Xaro Xohan Daxos was responsible for the kidnapping in the first place, she also finds out that Doreah is safe and sound and in his company.
6. Dany deduces that Doreah had sold her out to Xaro in order to keep herself safe and win the wealthy man's favor.
7. Doreah protests that she all did it to keep Dany safe, but at this point of time she only has Doreah's word that this is true, and Dany has no reason to believe her at that moment.

In short, the whole ordeal has made Dany a less trusting person (the lesson she thanks Xaro for), and now sees Doreah as a traitor who deserves death.

Did Doreah really do what she did to save Dany's life? Perhaps. But the price (the death of Irri and the kidnapping of the dragons) was too high to justify it in Dany's eyes.

Also, in a deleted scene it is revealed that Doreah herself killed Irri with a strangling cord when Irri walked in to find the dragons in cages, so it is truly a life for a life.

This should explain a bit more. This scene was deleted and adds more context.

Doreah betrays Dany by teaming up with Xaro, who in turn is teamed up with the Undead Warlock to steal the dragons. This is made clear by the bed scene - Xaro had asked Dany to marry him, then sleeps with her handmaiden (not very sincere, is he?) Also the masked mysterious woman from Xaro's party later tells Ser Jorah, when he seeks her out whilst looking for the dragons, that "the thief is with her now" (meaning with Dany), and in the scene prior she was last with Xaro.

As for the dragons, it is consistently emphasised all through S2 that whoever has dragons will have the ultimate power in the world of GOT. And that men will "lust" for that power. Xaro is the wealthiest man in Qarth; by elimination of all other motives, it's reasonable if not obvious to conclude Xaro bought Doreah off to assist with stealing the dragons because Dany wouldn't hurry up and say yes to marrying him.

And remember Doreah had no love for Irri - they are fighting in one of the scenes leading up to D's return to Xaro's villa where she finds her people dead and the dragons gone. Also, remember that Doreah was originally bought as a slave by Dany's brother for her in the first season. So they never started out as friends and yes even though Dany gave her her freedom later on, she obviously got greedy for Xaro's wealth and has either offered to help or had been seduced to help (although remember she was always a prostitute - she says so in season one, from the age of 12). She was always going to side with the highest bidder or whoever could offer her the most.

Well, let's see...

1. Daenerys's dragons are gone and her friend and handmaiden Irri is found dead.
2. She risks her life trying to find her babies (dragons).
3. Dany finds Doreah in bed with Xaro.

Dany sees Doreah as a common whore and not her friend at all in the end, so she locked her (and Xaro) up in that vault for good reason.

Think about it, if you were in Daenerys's place.