This achievement is horrifying to coordinate and is the most difficult in the game, despite its partner achievement "I Can Do Better" having the higher ratio.
What is a friend challenge and where does it come from?
To locate your multiplayer challenges, you can either:
a) open the main menu, idle the selection on Multiplayer, push right on the D-pad, and select Challenges, or
b) open the multiplayer menu and select Challenges.
The game will list here every available challenge you have based on the play history of you and everybody on your friends list, so you could have several Beat Your Friend or Beat Your Best challenges lined up here. Beat Your Best challenges generally only show up in the absence of a Beat Your Friend challenge, but they do sometimes show up concurrently. Every challenge that appears here counts toward "I Can Do Better" when completed.
The way a friend challenge surfaces is that someone on your friends list needs to beat one of your statistics (kills, objectives, stuns, etc.) then open his Challenges menu to send you a friend challenge. They do not simply appear on their own and require interaction on both ends (by your friend to challenge you, and by you to accept it).
What do I need to know that the game does not tell me?
Future Soldier dumps your challenge statistics once every 14 days (the stats that generate friend challenges, not the number you have completed or anything like that). Nobody is sure if this is based on play time (i.e. every 14 days starting from when you begin playing), if it's date based (i.e. the same day for everybody), or if it's based on inactivity (i.e. not playing for 14 days). Ergo, what generally happens is that a session host will (should) put up a session for it 2 weeks in advance and advise everyone who joins to simply not play for all the time leading up to the session. If you are in a rank boosting session on Monday and join a friend challenge session on Tuesday, what ends up happening is that your stats are severely bloated from boosting and you end up passing some ridiculous challenge over like 60 stuns in one match. This is very bad for many reasons, most notably that it would take the poor bastard paired with you the entire match to beat it, if it's possible at all. The short version is, do not play the game for a minimum of 14 days leading up to friend challenge sessions.
Similarly, the game will generate challenges from anybody on your friends list who has played the game within that 14 day window. You must either determine if your friends have been off the game long enough where they will not cause an issue, or you must delete them from your friends list for the duration of the session. Future Soldier is very iffy about how it compiles its challenge lists, and a lingering challenge from someone on your friends list could replace the one you want from the guy you're paired with in a session.
So how do I complete a volley?
To begin with, someone on your friends list must beat one of your stats (made much easier if your stats are all zeroed, obviously; "More than 0 kills in a match" is much better than "More than 45 kills in a match"). They will then visit their multiplayer challenges screen, where it will prompt them to send a friend challenge to you. You will then have a friend challenge with 24 hours to beat it. The pair of you must then simply beat the challenge and send it back to the other with the time limit halving each time (i.e. "volleying" the same challenge back and forth). 1 hour is the final window to complete a challenge, and the person who completes that gets the achievement.
How do I boost it?
Given that you can only do this with people on your friends list anyway, "boosting" is almost a misplaced term, but I digress.
The established method is to collect 8 people (the required number to fill a lobby), then squadding up in groups of 4 and searching for each other in squad matches. Squad matches are totally dead and you will never find randoms in there so it's perfect to work on something like this.
The only way the game allows a squad invite is for the squad leader to have the person they want to invite on their friends list. Whomever is setting up the squads must friend each of the three people who are squadding with them, then delete them from their friends list once they're all in the squad. Once the squads are formed, everyone must now pair off with someone on the opposite team by friending each other on Xbox Live. Now, both squads must simply search up a Saboteur match to find each other.
From this point on, I will refer to the squads as Squad A and Squad B. It does not particularly matter which squad actually ends up as Ghost or Bodark in the game, you must simply keep them straight as you continue boosting because at the end of each match, the teams are swapped. Because the teams and challenge completions should swap at the same rate (once per match), that means what should end up happening if you follow my advice is that one faction will always be dying or doing nothing. For example, if you begin with Bodark in the following paragraph, then Ghost squad should always be doing nothing. In addition, Squad B will receive their achievements first.
After the squads find each other in a squad match, each person on Squad A must open up their challenges menu from the lobby and accept what should only be a single Beat Your Best challenge. These should be "do better than zero" challenges, e.g. "More than 0 kills in a match", "Kill from more than 0 meters", etc. When the match starts, Squad A will do the absolute minimum to complete the challenge (e.g. 1 kill in the entire match, kill an enemy from point blank, etc.) and must double-check the Progression screen in the pause menu to make certain the challenge done. When Squad A is sure they're all done, someone must simply grab the bomb and plant it to end the match as fast as possible.
What the previous match has done is "prime" the friend volley, meaning that each person on Squad B should now have a 24-hour Beat Your Friend challenge for, hopefully, something really easy. They must accept their challenges before the lobby timer expires, then simply complete the challenge in that match. Squad B should then volley the challenge back where Squad A will have 12 hours to complete it. Squad B then has 6 hours, Squad A then has 3 hours, and finally, Squad B has 1 hour. Upon completion of this 1-hour challenge, Squad B gets their achievements.
Now is the fun part, since everyone had to fuck up their stats in the boosting process and you have likely burned at least 2 hours of a session already. What was once e.g. 0 kills in a round is now 7, so the entire process takes longer. Squad B now needs to complete a Beat Your Best from the Challenges menu because the previous volley exhausted the statistic in use for that challenge. Completion of this Beat Your Best should give Squad A a Beat Your Friend challenge with 24 hours on it, and the process must simply be repeated.
The long and short of this achievement is that whoever accepts the original friend challenge and has 24, 6, or 1 hour(s) to complete it receives the achievement in the end. Whoever receives the challenge first and has 12 or 3 hours to complete it does not get the achievement in that volley.
Good luck, you're going to need it.