I blundered through this conversation the first dozen or so times. Then I started to see common themes and found an effective way to get through it.
I usually use phrasing that doesn't carry any negative connotation, but establishes a bond between you:
"Can I share something personal about myself with you."
"I'm in the process of changing my gender" OR "I'm thinking about changing my gender."
I'm not a fan of letters. I told everyone I care about face to face. I arranged a place that is quiet where we wouldn't be interrupted, and left the time for the conversation open for as long as was necessary to answer all their questions.
Is it possible that your mum will come away from the conversation a little bit apprehensive, and will tell your dad? My parents are highly co-dependent and share everything with each other. I've found there is no point telling them at different times, as they always end up sharing with each other - and the conversation is much better coming from my own mouth since I can describe my experiences better to my father than my mother can second-hand. Something else to consider, anyway.
Good luck to you!