By road from London it takes about 2 hours. Take the A12 from London to Colchester; change on to the A120 until you reach the junction with the A133, which will take you via the 'Weeley Roundabout' to Clacton-on-Sea.
Clacton's railway station is about 15 mins walk from the seafront. There are direct trains every hour to/from London Liverpool Street. The journey takes approx 1and half hours. From other directions change at Colchester, which is half and hour away.
There are regular buses from Clacton which can take you to other small towns along the coast. Bus timetables available in the station concourse. For example, buses go half-hourly from Clacton to Frinton and Walton-on-the-Naze.
Like most of my other journeys round the South of England, I tend to take the train. Trains to Clacton take about one and a half hours, departing from Liverpool ~Street station in London. the service is not that great, only going every hour (at eight minutes past the hour) during peak times.
As always, please use the appended website to check details.