I haven't sailed to Calais with a bike, only to Dunkerque, but as for the Dover end:
When you reach the port entrance, follow the red line. You cross the entry lanes via a light-controlled crossing and then cycle up the right hand side of all the traffic to the French passport booth. After that you go through but don't follow the cars, still follow the red line which takes you on a short cut to the right where there is a separate building and you check in for the ferry there on foot. (This is for DFDS, P&O may differ). Then you get back on your bike and go to your lane, alongside the cars this time.
When it's time to board, you ride to the ferry but you have to push your bike over the threshold because of the grooves in the ramp. Then you will be directed where to put your bike.
Where your bike is stored on the ferry varies. I went out Dover-Dunkerque with DFDS at Easter, and on the way out all the bikes were lashed up together inside the bow. On the way back though there was only one other cyclist apart from me and we were told to put our bikes in a bike rack at the stern - outside and exposed to the elements. The bike rack wasn't even fastened to the floor and the weather suggested a rough crossing. I suggested to the other lad that we lock our bikes together and then to the nearby railing. This kept them upright but they did get a good dose of sea air and spray on the way across. No issues though except me being precious about my bike I suppose!
On the way off at Dover you are directed to a closed gate where you have to press a buzzer and someone turns up to let you through. Then it's follow the red line again to the port exit.
