You can do the trip in a day by bus, but it is a bit of a hassle as you have to get between bus stations in Bangkok. The Southern bus station, from Hua Hin, is way over the other side of town from the Eastern bus station, for buses to Trat.
A better option might be taking a public minibus from Hua Hin to Victory Monument in the centre of Bangkok,should only be a couple of hundred Baht or so, and then taking another minibus from there to Laem Ngop / Trat - these leave every hour. 300 Baht/person.
When you are in Hua Hin you might be able to buy tickets from a tour agent for a HH - Koh Chang minibus that would pick you up from your hotel . Most likely it would involve an early morning minibus to Pattaya - as there are a lot of these daily - and then changing o a minibus to Koh Chang.
Or of you want to avoid Bangkok city centre then a private car & driver would be the best bet but would cost way more. Looking at (I guess) 6,000 - 6,500 Baht total