North Kai2's walkthrough is helped, but I thought I'd elaborate on what I found worked.
Killionaire - 10 kills within 3 or 3.5 seconds of each other.
Perfection - win a slayer game without dying with 15 or more kills.
Triple Double - finish a game with double digit kills, assists, and headshots.
Steaktacular - Win a slayer game by 20 or more points.
Unfrigginbelievable - a spree of 40 kills.
Extermination - wipe out an enemy team with at least on overkill.
Killionaires, perfections, unfigginbelievables and exterminations aren't even worth going for. They're too difficult and spotty.
I did get 3 or so exterminations playing Halo:CE in the Team Slayer playlist. The teams are 4v4 unlike the CE playlist, and the grenades are so strong and spawns so bad you can get lucky just chucking grenades.
What you want are steaktaculars and triple-doubles.
Steaktaculars I got somewhat consistently on the H2A and CE playlists. Those seem to have high dropout rates, which help your team run up the score. They also seem to either steamroll one team or the other. That's all luck of the team assigned as you search, but every other game it seemed I would get a great team versus a terrible team, or I'd be on the terrible team (being on the terrible team allowed some killjoy manufacturing as you can feed kills). Not reliable, but I was getting maybe 1-2 or 2-3 every 20-30 games.
In the CE playlist, it's 2v2. Basically one bad teammate can sink a game. Hang-em High, just where out the active camo and the rockets and you'll run up the score no time.
---a tip: I play somewhat aggressively, I get high kill counts, but also get killed a lot. There were a few games where I purposefully hid and stayed back late in the game to get steaks by taking my deaths away from the other team. Some games come down to the bubble, you'll be around 17-18 up. Sometimes hiding can work
These are your best bet and most reliable to manufacture. Everything else revolves around luck or absurd skill that is beyond most. These you can actively work towards.
10+ kills, 10+ headshots, 10+ assists in one game.
Halo 4 is by far the easiest to get assists in, so I suggest that playlist.
Remember, you get loadouts in most gamemodes in this list. So take your time and make one. A BR and a plasma pistol, regular grenades, a jetpack, dexterity and shielding for me.
You'll get the hang of how much damage you to have to do to get an assist. 2 bursts with a BR will get you one almost always. DMRs work, just not as well for me personally.
Charged plasma pistol shots that take down shields and someone else kills them DO count. I have verified that. It's not a regular assist that comes up, but it DOES count. Similarly "Headcase" does count towards headshots even though it isn't a headshot medal. You can use the sniper or beam rifle, but they're a little harder to judge whether or not you got the headshot or not in game. So you'll have to be sure.
If you see a melee fight, or an assassination, put a couple rounds into the enemy. You will get an assist. Guy on your team with a sword about to lunge? Take a shot, you'll get an assist.
There are three gamemode slots to choose in the lobby. Usually an objective type: CTF, oddball, extraction, dominion light, king of the hill. Regular Halo 4 Team Slayer, and Legendary Slayer.
NEVER choose legendary. It has less kills, no loadouts and is just a little tougher to get assists. You die quicker, so it's harder to trade a kill for a death as well.
You usually want the objective gametype. You're not fighting over a limited amount of kills to get your 10 and 10 (a third of the kills have to have your handiwork). Extraction can go too fast sometimes if one team is too good. King of the Hill is almost ALWAYS a triple double for me as it can take so long you have so many chances and you know where your team will be to help with the assists. (Dominion Light assists for building turrets or capturing bases do NOT count. Hill assists in KotH DO count). You'll have to be a little more aware in those gamemodes.
Unfortunately nothing can help your aim and skill besides practice. You simply have to get headshots. Hang around objectives, or just out of the zone, use your radar, be aware of teammates, know that people go for the powerweapons. The jetpack can help you escape or get the drop on someone. If you know someone is coming at you, or close by, gaining altitude quick or jumping up a level when someone thought they were safe can help you get a couple of shots of first.
What I try to do is count out loud my headshots and assists. "0-1....okay 1-1.....2-no 3-1" until I'm pretty sure I'm at 10 headshots and just start trying to get assists. You will lose track some, so I will get another 1-2 if I'm unsure. But counting helps. Just follow a teammate around and throw some fire into the next enemy but don't go for the kill. Watch for that +5 points pop up on your screen after death if you did damage on an enemy. Congrats, your teammate cleaned up your garbage and you got the assist.
Hopefully that gives you some tips or help! Good luck.