Hi,
"I am writing to you to express my great desire for successfully performing the...""I am writing to you to express you my great desire for successfully performing the..."Which is the good alternative? thanks."I am writing to you to express you my great desire of successfully performing the..."
The second and third phrases are clearly wrong. The first phrase is somewhat stilted but acceptable. Assuming you have met the person, I would leave out "to you" because it is obvious from the context of the sentence.
Try this: "I am writing to express my great desire of successfully performing the..."
Or this: "I am writing to express my great desire to successfully perform the..." if the task has not yet been performed.