I've recently helped a client with this process, and in my experience it wasn't that bad at all.
We simply sent out our trademark registration information (publicly available from USPTO), and all the other required information from twitter policy. It took us about 3 weeks to regain the twitter handle, and most of the e-mail communications were automatically generated from twitter's e-mail bots.
Here's my checklist of things:
1) TM registration number. If you don't have one, probably having your TM application will be a good start here.
2) You need to show a bona fiderelationship between the handle in question, and yourbusiness/trademarkable name. Factors that help here:
a. Domain ownership.
b. Name of the business/DBA.
c. How long have you been using the name for; the longer the better.
d. Being the senior user of the mark, i.e. you're the first to use the name/mark.
e. Any evidence of brand recognition in the marketplace. For example, you can point out that the first few google page results of the name in question points directly to your product/services. Social media presence helps a lot here too.
f. Potential harm to yourbusiness if the handle stays with the current owner: this is far more relevant if the handle owner actively tweets things that may create consumer confusion in the marketplace. If this is the case, you should make note of the damaging tweets and attach them as a pdf printout in your request (if possible; I never had to go through this far)
Let me know if you're interested for a courtesy consultation, and I can help you with the TM process and/or the transfer request. At the very least, I can share you some of the document templates that might save you time and frustration in generating these kind of letters.
Best of luck.