If you want to travel independently try the luxury train Exporail. https://onlinebooking.exporail.lk. Leaves Colombo at 9 pm and arrives in Trinco at 5 am. Bookings are still not available for July though.
Or you can hire a vehicle with a driver from one of the reputed companies if you wish to stop enroute to enjoy some other sights. One suggestion is to witness large herds of elephants in Minneriya (this is the peak season for 'The Gathering', which is billed as the world's 6th greatest Animal spectacle. ) If Trinco is your only destination it maybe best to travel back independently. Or since it's probably your first visit to the East coast (and you are traveling during the season -weather wise for this part of the island) you should probably enjoy this unspoilt coastal stretch by visiting Passikudah, Batticaloa and Arugam Bay as well.
Enjoy your eastern coast vacation!