Part of Heathcliff's passionate nature is his raving "like a mad dog," as Catherine has said in the novel. Heathcliff both inflicts pain and requests it (as he wishes for Catherine to possess him in the end). His mood swings from one intense feeling to another. A letter from him should be nearly psychotic.
In addition to mkcapen1's points, I would add something about the destructive nature of the relationship of Heathcliff and Catherine. Remember, their relationship can never be described as a "normal" romantic relationship, and seems more based on mutual self-hurt and self-harm than anything else we would rather see in a relationship. Therefore, at times, Heathcliff would rage against Catherine for how he has been hurt by her actions, what she said about him in Chapter 9 when he overheard her say that it would "degrade" her to marry him, and then her eventual choice of Edgar. Get passionate and angry!
If I were to write a letter as Heathcliff I would add certain facts such as the following facts.
I have loved you since childhood.
Together we ruled our world.
Your father gave me to you and we are meant to be as one.
After our father died Hindley took everything for me.
He made me a servant in our home.
We stayed together every chance that we had.
You loved no other.
Later you stayed with the Lintons.
They kept me away from you and I was crazy with worry.
You forgot who you were, who we were, after your stay with them. You became materialistic.
You still loved me but you married him.
I returned to you a better man. I was able to take you away and give you the things that you wanted, bu you stayed with him.
You still loved me, but you had his child and it killed you.
I have and always live only for you. Forever.