I think train is probably your best bet. A quick google shows it's a single train line from London (Liverpool train station) to Colchester so there's really not much more you can ask for. Cab / uber from the airport to the the Liverpool train station and should be smooth sailing from there.
As mentioned, there are a few airports so maybe worth flying into the one that's closest to Liverpool train station as cabs can be expensive. To give you an idea of cab cost, i paid 90 pounds (was around $180 CAD at the time) from heathrow to downtown London a few years ago. Heathrow might be the closest to that train station based on my limited memory of the city.
re: flights, i'm personally a fan of google flights. you can save routes/flights and it will email you when the prices change.
www.google.ca/flights
EDIT: don't listen to me, tony's National Express Coach suggestion below seems to be a better option vs. cab+train