This can be done in a private match if you have enough people. To earn legitimately, It’s a very tricky trophy, made harder by how unstable the multiplayer mode often is. But it’s doable.
To clarify what needs to be done, as Batman or Robin you must take out all six enemy players, using a different takedown on every one of them. As such you can only do this if both sides have a full team of three people each.
The way this trophy tracks your progress works like this. When you take someone down, that particular takedown is locked in on that particular player for the rest of the match.
For example, if you used a silent takedown on Player A, both the silent takedown and Player A are now counted for this trophy's progress until the match ends. If you later take down someone else with a silent takedown, you can't then takedown Player A again with a new move and have it count, because the game considers that player already counted.
As such you need to keep going to different players and keep using different takedowns. Dying won't reset your progress and you don't have to do exactly six takedowns on six different people. Reusing a takedown, or taking the same player down again won't mess up this trophy (as far as I know) it just won't add any progress to unlocking it. As long as at some point you've taken out every enemy player using a different takedown from the previous ones in the same match, you'll be fine.
There are several takedowns and what counts as different ones can be a little confusing. Here's the ones I know of.
Inverted - Done from vantage points.
Silent - Hit Triangle while standing behind someone.
Smash - Hit Square while doing silent takedown.
Ground - Done while enemy is lying on ground.
Grate - Done from inside of grates.
Ledge- Hang from ledge and wait for enemy to pass by.
Corner- Take cover behind wall, wait for enemy.
Double - Hit triangle while behind two enemies near each other.
Reverse Chokehold - Done from edge of roof with enemy below.
Melee - Hit enemy (square) until beaten. (Usually takes 3 hits)
Gadget - Take out enemy with explosive gel or sonic batarang.
Both explosive gel and the exploding batarang count as the same takedown for some odd reason. So only one of them count towards this trophy. The double takedown can be troublesome. Only one of the two taken down count for it, and I'm not sure which. I assume whoever's name appears first on the kill tags, but I'd just avoid that one, which is easy enough considering how rare the chance to do it is.
I found the six easiest takedowns to do are Inverted, Gel, Silent, Smash, Ground and Melee. Only the first one requires any kind of special environment (someone must walk under a vantage point), the other five can be done anytime.
Start with the Inverted Takedown to get it out of the way. Try and work in the ground takedown whenever you see someone isolated from all the other players. For the explosive gel takedown, glide kick or drop attack an enemy, then quick fire the gel (L2 + Square) then hit quick fire again to detonate. As long as you set it off before they get up, they'll get KOed.
The last three are all pretty simple. Silent and Smash just involve getting behind someone. Don't bother sneaking like in the single player, running doesn't alert human players (but gliding can so try to drop straight down without gliding). The melee one just involves hitting someone. Try to do it from behind though. If the catch any glimpse of you first there's a good chance they'll kill you before you land all three blows.
Robin in particular is good at attacking minions. The zip kick (L2 + Triangle) immediately closes the gap on a minion and knocks him to the ground, making him an easy kill for both the explosive gel and the ground takedown. He also has a very durable bullet shield you can use to block attacks until you're close enough to bash someone. Just watch out for grenades as the shield won't stop them.
The trickiest part of this trophy is keeping track of which players you've knocked out. I suggest keeping a pen and paper handy. When you take one out, write down who and with what take out. I know it sounds OCD, but with all the odd PSN names out there it's really hard to do this from just memory.
It also helps to focus on one team first. Jokers are red on detective vision, Bane is blue. If you take out a Joker first, try to get the other two Jokers next. If you can get all three with different takedowns, then you know the only ones left you need are Bane's people, or vice versa.
Team captains have their names and a white star above their heads, but otherwise the game doesn't tell which player is which until after you KO them so there might be some trial in error locating certain players, but remember their clothes and weapons are different, so turning off detective mode can be handy for telling them apart.
You don't have to do the takedowns I recommend. If the opportunity presents itself to do other ones, feel free to take it. And above all be patient, you don't have to win the match, you're just trying to get this trophy. So don't worry about getting KO's to win. And if you're close to winning and haven't gotten the trophy yet, you can let someone kill you to drop your score a little. That way you don't win by accident before the trophy unlocks.
If one side is dominating the other, you can slow them down the by sabotaging their control points with scramblers. Just walk up to the computer panel at the control point and hit X. Also feel free to use smoke bombs, batarangs and the exploding rang to muck them up. Just getting hit by these tend to slow people down because they figure you'll be attacking soon.
I earned this trophy on Joker’s Funhouse, which is probably the easiest map to get this on. The maze like set-up often causes people to get split up, making it easier to pick them off. The vantage points are also spread out all over the map instead of just on the sides, making it easier to get around and harder for other players to anticipate you. Wayne Chemical can work due to the fairly open ceiling area. I recommend avoiding Wonder Land and especially Blackgate if possible. These maps have a lot of tunnels where’s it harder to reach people.
With some luck, you should be able to get it. Just take it one guy at a time and know which takedown you want to do next.