An invaluable tool for this is the Instance Achievement Tracker addon. I strongly recommend this addon if you are trying for achievements.
Here are some thoughts on the achievements we did:
Every single person in the raid already had the achievement to destroy 10 spikes in 3 seconds. Judging by the dates of the achievements, we actually meet the requirements are every week in the regular raid.
This was fairly easy. Just clear the trash before the boss. Have a tank run out and get the Disciple and drag it to the boss. Make sure everyone is in the raid is inside, pull the boss, and wait for the Disciple to split into 3 before attacking it.
We had our second tank hold an add in one of pools until the stoneskin phase. Then the tank got an orb to hit the add, we killed the add, and then finished attacking the boss.
This one took us two tries. The first time, the orb accidentally hit the tank instead, and the add did some sort of self-sacrifice move before we could get it to hit a second orb.
This one was the hardest of the ones we did. You have to knock the ball into the goal in the last 3 seconds before the debuff expires, and do this 3 times in one fight. We had two people bounce the ball back and forth in front of the goal, and then they or the third person knocked it in at the 4-5 second mark.
The biggest problem here is that there is no feedback if you are successful or not. Blizzard should have had a chat line on a success or maybe have the fireworks that go off if you score be a different colour.
The one saving grace is the enrage timer on Xanesh is really long. We ended up doing about 6 ball phases, just to make sure we got three successes. There was a whole bunch of "Did that go in on time? I think it did, but I'm not sure." After the second ball phase, we had to stop dps entirely. The tank did get one-shot at some point, but I think we could have avoided that with better tank swaps.
Everyone in the raid needs to get hit with 30 stacks of the beam debuff. A lot of the guides emphasize stacking and using raid cooldowns. I think it's easier to just take your time, and hit everyone with a few stacks every beam, while doing other mechanics properly and avoiding death.
IAT is a lifesaver here, as it notifies you when people reach the limit and who still needs stacks.
The plan is to do a few more achievements tomorrow and finish up over the next couple of weeks.