From LCY, catch the DLR (light railway) to its terminus at Bank. Try and sit at the very front of the train (if the next train is 5 or more minutes away, get the train going the opposite way if you want to nab the front seats!)

From there, you can either change for the Central Line to Liverpool Street, or just walk (about 10 minutes).

To get to Heathrow from there, well, that depends on your preference.

You can get the Circle or H&C line from Liverpool Street to Paddington, then catch the Heathrow Express or Connect from there. This is probably the least reliable Tube line.

Alternatively, you can catch the Central Line to Lancaster Gate and walk 5 minutes to Paddington. This is faster, but you do need to know the way.

If you're going to get the Heathrow Connect, then you may as well catch the Central Line to the Ealing Broadway terminus and change there.

Finally, if you want to go cheaply, you can get the Piccadilly Line all the way there (changing at Holborn). This will take over an hour.