Reach character level 50. This will be quite the grind, but Clan Battles award generous amounts of karma towards this. A decent match can net upwards of 100,000 karma. My best match so far is around 250,000. Just keep at it and this will come with time. While you could grind this out in offline, single player Trials, I do not recommend it, as the karma awarded is signficantly lower than MP.
Boosting method, courtesy of cra:
If you have at least 1 friend who is willing to boost you have a faster way than grinding one trial over and over again.
Map: Doesn't matter as long as you know the spawn areas
Outfit: Choose the same color for an extra 500XP bonus per kill and either the same headgear or armor for an extra 1000XP bonus per kill
Now for the killing part. If you are really lucky, you will start with the "Life of a Killer" contract, which is the best possible beginning contract for two reasons. First, this contract doesn't change until you die (which shouldn't happen in a boosting session) and second, it gives 5000XP instead of 3000XP like the other contracts. The worst case scenario is getting a Falcon Dive contract, as it is time consuming and annoying.
As soon as the match starts, the Killer runs across the map to his victim and starts killing and fulfilling contracts. If you don't start with "Life of a Killer" you may receive it after a few kills. The fastest method for a kill is a head shot as long as you have arrows; after you run out switch to ghost kills and if it is available always do an UT kill for an extra 1000XP steel on bone bonus.
The best result was something around 395k in 5min, but it definitely averages above 200k per match/5min if you do it 1 on 1 with 27-32kills in the time limit. If there are more participants the XP earned will increase accordingly.