Check to see if there is any additional paperwork you have to fill out to obtain a travel permit. The department giving out the travel permit should be able to provide paperwork to you, if any is necessary.
Address the letter to the proper person or simply say, "To Whom It May Concern:".
Explain who you are. Write down your name, address, age and citizenship status.
Write down information about the purposes of your trip. If it's pleasure, say so. If you're doing research, indicate the research. The more information you give, the more able the person issuing the travel permit is to allow your request.
Write down any other information about you trip ---- whether you're taking safety precautions, how long you plan to stay, what vehicle you'll be using to travel and whether anyone is accompanying you.
Thank the person for their time and ask that a response is given by a certain date so you can make plans. Attach the paper to any additional paperwork you need to complete and send it to the department that issues travel permits.