TL;DR: You can get the Flawless Raider Achievement/Trophy on your 2nd, 3rd, or xth time through a raid on the same character in the same week, but only if you never died in that raid on that character any of the previous times in the week.
So, I carried a friend through Crota's End for the Flawless Raider Achievement/Trophy this past weekend and we screwed up three times - once on each character. After that we went back to our primary characters and tried again since we were so close. On our 5th or 6th run we had an absolutely perfect run and neither of us died. My buddy got the Achievement but I didn't.
I started thinking about it and the only difference was that I had died on my first playthrough on that character and he hadn't since I was having him stay at the spawn point and soloing the raid for most of the checkpoints. He hadn't died because when I did I took us back to orbit and we restarted the raid from the beginning. So, even though I ran the entire raid flawlessly and never died I didn't get the Achievement because I had died on that character earlier in the week in that raid.
I feel like we perfected the raid though, so here is some advice to others (particularly if you don't want to solo it or if you are trying to help another guardian get their flawless raider Achievemnt/Trophy):
First off choose easy mode, there is no reason to attempt flawless on hard mode.
1. The Abyss - Striker Titan with Mida Multi-Tool, high impact sniper, and heavy machinegun, max armor and agility, increased height jump, and shoulder charge. Run past the first two lamps and jump on the rock behind and to the right of the 2nd lamp. Wait for your weight of darkness to disappear and double jump shoulder charge over the wall (this is super easy with Mida and you won't even need the shoulder charge). If running with two players either have one wait at the spawn point or make sure to jump off of the rock one at a time so you don't bump into each other. Run to the end and open the bridge (you'll only have to kill one knight on your way there). Run past the last lamp and activate the bridge. Use the last lamp to jump onto a rock. Kill the first ogre which shows up and nothing else (more ogres will spawn if you kill more enemies). Run to the light once the bridge is fully opened.
2. Bridge the Chasm - Striker Titan with Mida Multi-Tool, high impact sniper, and heavy machinegun, max armor and agility, increased height jump, and shoulder charge. Trigger the encounter and jump onto a column. Kill the sword knight and jump across the chasm. To do this jump on the column to the right of the center platform (first one of the upper platform) and double jump towards the invisible center of the bridge. Only start slashing once you are at the peak of your jump and then turn towards the top of the bridge after you get to the point where you are over the bridge. If you are unfamiliar with this jump, practice it many times before attempting the flawless run (I was able to get my buddy across because I have 2 Xbox Ones and I hid while he handed me his controller to make the jump). Kill the blue gatekeeper and hide half way up the right path on the middle raised section. Kill everything that attacks you and watch for the right ogre. Kill him as soon as you can and then take your time sniping the other ogre.
3. Shrieker Hallway - Blessing of Light Defender Titan with high impact Hand Cannon or Scout Rifle (firefly really helps), high impact sniper, and Sleeper Simulant, max armor and agility, and increased control jump. You won't actually need the Sleeper Simulant here, but you will want it on the next checkpoint and odds are some of the thrall at this checkpoint will drop heavy ammo. Take it nice and slow. The shriekers do not respawn on normal mode. Kill the first shrieker and back up to avoid it's blast. Start slowly inching your way up the hallway killing everything (a primary with firefly really helps here). Kill the second shrieker when it opens and run and jump through the rest of the hallway. Use your super if you take too much damage. Make sure to feather your jump as you fall into the last room so you don't die.
4. The Deathsinger - Blessing of Light Defender Titan with high impact Hand Cannon or Scout Rifle (firefly really helps), high impact sniper, and Sleeper Simulant, max armor and agility, and increased control jump. Skip the first two knights and jump up to the ledge above. Run towards the door and wait for the boomer knight to come out. Hit him with one shot of Sleeper Simulant and finish him off with a bullet from your primary (if you don't kill this knight he constantly shoots at you the whole time you are engaged with the wizard and shieker). Run into the first round room and wait for the wizard to charge you. Kill her with two shots from Sleeper Simulant and hit the shieker with one shot of Sleeper Simulant as soon as it opens. If solo, repeat process on the other side of the room. After you pop the shieker run out and hide in the upper rooms to the left and right of where Crota spawns and kills some ads. Push in and hit the Deathsinger with 2 shots of Sleeper Simulant when ads are clear. Return to the safe spot and kill ads until the checkpoint triggers.
5. Crota - Blessing of Light Defender Titan with Touch of Malice/Red Death, high impact sniper, and heavy machinegun, max armor and agility, and increased control jump. Touch of Malice is a great way to bring down Crota as long as you have blessing of light, and Red Death is a great way to get your health back if you're injured. Alternate between them. Play it safe and hit Crota twice every time he's downed (It only takes 6 hits - slams not slashes). Two quick rotations and you're done.
Good luck getting your Flawless Raider Achievement/Trophy guardians!
[b]EDIT:[/b] Many redditors have informed me that this isn't true and that my Achievement should unlock over time.
-
If you die, reset the raid and do it flawlessly, you'll get the trophy. I did it by myself on one character. It took me a few tries, but I still got the trophy when I was done.