To revive a downed teammate, you need to tap X while standing near them. They'll stand up instantly with full health, if completed successfully.
The leaderboards are not a good representation of your progress toward the achievement. It's glitched, so it will count revives that the achievement does not. Such as during lag and after players quit.
You need at least six players present at the end of the game for your revives to count. If too many players quit, you will not gain progress toward the achievement.
It will take several games to do this without boosting; possibly taking from days to weeks to achieve, but if you insist on doing it legitimately, Gridlock or Clocktower are good choices for lots of revives. Just stick close to your teammates and be ready.
Personally, I'd recommend boosting. It will take far less time and much less effort. To boost this you'll need at least six people, as you can't start a ranked game with less than six.
Start a versus ranked match with the following settings:
Game Type: Warzone
Selected Map: War Machine
Number of Rounds: 19
Bleed-Out Duration: 60
Round Duration: 10
Have everyone search for the match and join. Start the game when everyone is in the lobby.
When the game starts create two lines across from each other, COG in one line and Locust in the other, then have the first person in front of the COG line shoot the first person in the Locust line with a Lancer until down. The second person in the Locust line will revive him, and the COG will down him again.
After the second person revives him again, he will go to the end of the line, and the new first and second person in line will repeat what the previous did. Do this until everyone on Locust team gets two revives.
Once everyone on Locust has had a turn, COG will kill all but one of them. That person will then down the COG as the COG gets revives just as the Locust did.
After everyone on COG has two revives, have one team kill the other to finish the round. Make sure trade off round wins to make the match last longer.
Using this method, you'll play 37 rounds with two revives per round for 74 revives in one match for each person. If someone doesn't get their achievement in one match, you can play another five to ten rounds match. It shouldn't take longer than two hours to get all 100 revives.
I learned this method from
https://www.trueachievements.com/ExtremePhobia.htm. We did it in a boosting session together, and I can verify that it works.