There are two ways for hitchhikers to cross the English Channel: by ferry or by the Channel Tunnel. Ferries leave from Dover on the English side and either Calais or Dunkerque on the French side, and the tunnel runs from Calais to Folkestone.
Both the ferries and tunnel are free for hitchhikers, as drivers pay per car rather than per passenger. Most drivers purchase tickets for the tunnel and the ferry in advance. It is very important to tell drivers that it's free to add passengers to their ticket, as most drivers don't know this! If they say that they had to list the passengers on their ticket, tell them that it's very easy and free to add another passenger at the ferry terminal/tunnel entrance.
It is highly recommended to find a ride across the channel well before the border and cross by whichever means your driver is going. There are a few spots where hitchhikers have been highly successful in finding rides across the channel:
From England To France
You're best off finding a ride at either Maidstone Services on the M20 or Medway Services on the M2. Even if you're offered a ride to Folkestone or Dover, it's often best to turn down these rides and wait for a ride going all the way to France.
If you get tired of hitchhiking from one of these service stations, or are beginning from a spot further southeast, then it is often easier to hitchhike a ferry from Dover than getting a ride through the Channel Tunnel in Folkestone (although both are possible).
Personal experience, June 2019 I got to Maidstone services and immediately found a van with a Berlin number plate. They couldn't take me, though, because - as most people do - they had booked their ticket in advance and had to state which and how many passengers would be in the car. (If they go for the cheapest option, they cannot amend their ticket.) What worked instead was asking a group of Polish freight drivers in front of their vans. Most of them have a ticket with P&O Ferries and don't need to state how many passengers are in the front. No questions were asked by the ferry/ticket operators. So, as of today, go for the commercial drivers instead of private people.
Personal experience, October 2019 We (1 male, 1 female) went to the Maidstone service station, approaching lorries to get a lift to France. A British driver told us since Feb 2019 most lorries can't take an additional "driver" aka hitchhiker for free and are not taking hitchhikers anymore. Interestingly Romanian and Polish trucks are excluded from this regulation since they usually drive a lot further and are expected to have a second or even third driver in the cabin. So if you want to hitch with a lorry, go for the PL and ROM plates. The British driver also told us you can't add extra passengers to the tunnel ticket so we'd do better to ask ferry passengers. We then phoned the two biggest ferry companies to check and yes - on most tickets you can add additional passengers without any fee. At Maidstone itself we couldn't find anyone going to Dover, so we went to the petrol station close to Folkeston and found a guy who took us all the way to Hamburg!
If going for the Dover ferry option, stand at the BP petrol station on the road before the ferry port (51°07'06.1"N 1°18'25.0"E) on Limekiln St/A20 roughly a mile before the ferry entrance. Ferries go to Calais and Dunkerque. If you're heading south towards Paris, take the Calais ferry and you'll likely find a direct ride. For those headed North/East to Belgium, Germany, and beyond, both ferries work but Dunkerque may be a bit better (Calais ferries go more often though). Accept any ride going on the ferry, as it should be easy to find a ride further towards your destination by talking to people on the ferry. See the Dover page for more details.
If going for the tunnel option, stand by the viaduct at exit 11a, where there is a big space for cars to stop. More information at the Channel Tunnel page.
From France To England
Look for a ride going to England from the last service station before Calais in whichever direction you're coming from. When coming from Belgium, the last service station before the French border seems to be a decent spot. Even if there is little traffic in the middle of the night, a decent proportion of the cars will be going to England.
If this fails, you'll most likely have an easier time hitchhiking from the ferry entrance than from the tunnel entrance. Some hitchhikers have reported being kicked out of the tunnel entrance by the authorities, while others have been successful. For ferries, there are two options: Calais or Dunkerque. The Calais ferry has more traffic and has the option of buying a foot passenger ticket if you're in a hurry and don't have a ride. The Dunkerque ferry is generally more laid back but has less traffic and no option for foot passengers, meaning you can only get on if you get a ride.
For either ferry, the parking lot near the ticket office is a good place to start asking for rides. Unfortunately, most people buy their tickets in advance and won't stop here, though, so if you don't find someone buying a ticket, then thumb on the road before the ticket booths/passport control where all cars going to the ferry will see you. Once past passport/ticket control, you can ask cars where they're going and try to find a ride further in your direction in England. If you get a lot of rude denials, don't give up! You'll find a ride! See the Calais and Dunkerque pages for more info.
For the tunnel, there is no great spot to stand, so look around for a discreet spot where cars can stop. Make sure you're not in view of the police or tunnel employees. Hitchhikers have been successful at the motorway entrance as well as standing behind the arrow sign where the two roads split before the tunnel control. It's recommended to ask your driver to not reveal that you are a hitchhiker to passport/ticket control, as there have been reports of hitchhikers being denied entrance to the tunnel. More information at Channel Tunnel.