Sisterhood: Play 3 matches on the same team as Ambra.
Wait until other people have chosen their characters and then choose Galilea if someone has chosen Ambra. Otherwise use a secondary account that has her unlocked and play hardcore splitscreen story missions and suicide instantly to get this done in 10 minutes. (Ambra unlocks at command level 14 or by winning 5 matches as a Jennerit)
Frenemies: Participate in killing Ambra 5 times.
Get a kill or assist on an Ambra in PvP. Either boost this in a session or hope that you find an Ambra in a game and make it your sole mission to take her down. Be aware she has a passive that triggers when she is near death that grants her an overshield and possibly a speed boost, allowing her to run away. Saving your stun until she is nearly dead may be the best strategy.
Dark Knight of the Soul: Regenerate 10,000 damage with Abyssal Form.
You regenerate HP while using your ultimate ability (Y), this has to be actual health healed so use this every time it is off cooldown and you have some health missing.
Arbitration: Block 25,000 damage with the Sentinel’s Greatshield.
The shield is used by pressing LT and can absorb a certain amount of damage before temporarily breaking, similar to Boldur's or ISIC's LT shield. This should come in natural play but if you want to grind it an effective way is to block a bosses attacks, hide while it regens then rinse and repeat. The varelsi boss on the Void's Edge level is particularly good for this as it has strong ranged attacks and if you run to the entrance of the boss fight it cannot target you and slowly sends minions at you.
No Redemption: Kill 100 Jennerit enemies while Galilea is corrupted.
Corruption is Galilea's passive that builds up by attacking or using skills, it grants a fairly weak aura that damages enemies and steals HP. This does not require you to be at full corruption and is easily gained in one PvE story mission, such as the Renegade or the Archive which heavily feature Jennerit enemies.