I created a spreadsheet to help me track which matches I won and to also see what was coming up next so I could plan my Burn Cards accordingly.
The link below is the file on my public OneDrive (previously SkyDrive) account. I won't be removing the file. It lists each campaign match name and type. It has two columns for IMC and Militia. Each match has a 0 in each of the columns. When you win a match, just change the 0 to a 1 and the tally at the bottom will tell you the number of wins you still need. There are 9 matches in the campaign.http://1drv.ms/1fUNBaR
(You can click File / Save As to download a local copy to your machine)
The 9 matches you need to complete are as follows:
The Refueling Raid - Hardpoint Domination
The Colony - Attrition
The Odyssey - Attrition
Get Barker - Attrition
Assault on the Sentinel - Attrition
Here Be Dragons - Hardpoint Domination
The Three Towers - Attrition
The Battle of Demeter - Hardpoint Domination
Made Men - Hardpoint Domination
You do not need to complete them in one sitting. Doing a Quick Match seems to be best way to get the matches you still need. Doing this will most likely be quicker than trying to join a particular level. It was for me anyway. BlinkFandangoII's solution has information on things to try on that front. I had to do quick matches to get my Militia achievement knocked out after getting the IMC one.
If you do a quick match and have the rare opportunity to join the match close to the end and get the "Victory" message, that counts even if you didn't play the entire round.
Some people have said this wasn't the case for them. I thought this was the case for me, but it will be good to mark those "joined late" to redo if the achievement doesn't pop for you.
Also, look at the comments to see that some folks had to exit Titanfall and come back in for the achievement to register.
Hope this helps
--Update from Leo Ascendent in Comments:
New update puts a little trophy icon next to the faction you've won the campaign mission on. Might want to update with that info, pretty helpful in finding what you need to do still.