UPDATE: As Faustus PHD writes in his solution, private servers have now been added to the game and this achievement becomes therefore boostable. I'll keep my solution online for people who still want to get the achievement legit.
I did this achievement in derby-mode only, so my solution is focusing only on doing it there.
About the requirements: As others already said, you need to have at least one other human player in the game for a win to count - wins against a full lobby of AI-players don't count. You can recognize AI-players by the small green star in front of their name and the ping of 1.
Since there are currently no private matches you need to get your 20 wins on public servers. You can do this achievement on public servers legit or boost it with a partner in private matches. If you take the boosting-route you don't need any of my following advice. Another thing: The achievement-tracker doesn't work for this game, so you have to keep track of your wins yourself.
Preparation: First of all, I strongly recommend doing the grind to level 50 first, mainly for all the credits that you get along the way. So check the other solutions for "Showing Some Dedication" first and get to work on this. Once done go to the market and buy yourself a Gatecrasher and buy all upgrades for it, including all armor-upgrades. The Gatecrasher is in my opinion the best derby-car in the game by far. I used the Muddiger first which is a fine car too, but the Gatecrasher is slightly stronger and far superior in cornering and has better grip.
Choose your server: Under "Multiplayer" you can choose derby-races or check the server-list for derby-servers. Checking the servers also shows the amount of players currently playing. This is interesting if you feel more comfortable in smaller/larger-lobbies, for me smaller lobbies worked better. Keep the server-list in mind if you are stuck in a lobby with players who are too good and know what they are doing ;)
How the scoreboard works: Ok, the game is called Wreckfest and not Damagefest for a reason: Wrecking cars is your priority, not necessarily damaging them. The ratio of wrecked cars and how often you've been wrecked determines your position on the scoreboard. The damage dealt is only the tie-breaker, meaning if two players have a 3:1-ratio the player with the higher damage value is first.
Tuning your car: Change the transmission to "short". This will reduce your top-speed, but gives you better acceleration, which is more important in derby due to the small size of the tracks and the constant crashing.
Strategy for winning: Because of how scoring works I strongly recommend not to focus on damaging cars, but on wrecking them. Example: If you see a car with low damage it might be a bad idea to hit it if you already know that you won't wreck it. You only reduce it's health and prepare the kill for another player, putting yourself to a disadvantage. Always choose your targets and focus on low-health-cars. Also try to stay out of the all-out-carnage in the beginning of a match, because there is nothing to win here. After about half a minute into the game it starts to become interesting, since there will be quite a few cars with low health to pick off. Another tip is to check the cars in the lobby that other players and the AI choose. Your fully-upgraded Gatecrasher is an A-level car, if you see C-level cars you can sometimes reduce their health by two bars with a speedy hit. About speed: Don't drive in circles desperately trying to find a car to wreck. Even if you do, you'll be too slow to damage it properly. Always use the whole track to gain speed. This means sometimes to even leave the area where the most action currently is. Stick to this strategy consistently.