Which Order Do You Play The ME3 DLCs In?

From Ashes: If I choose to do it (sometimes I don't), I do it early on--after obtaining Garrus and doing the first N7 mission.

Omega: After the coup attempt on the Citadel. It makes sense to me for it to take place after that.

Leviathan: Before the main mission with the quarians.

Citadel: After the main campaign or before Chronos. It depends on my mood.

From Ashes - I will play usually play this one very early in the game to get Javik right away. If I'm really interested in taking Ashley, Kaidan or Tali on it, I'll do it right after the Citadel Coup.

Omega - I usually do this one before the Citadel Coup right after completing her Citadel assignments.

Leviathan - I usually do it right after Thessia.

Citadel - I usually do the entire DLC, including the party, just before Chronos. I've honestly never tried doing the combat portion early and then doing the meetups with the crew and party later because I honestly thought that if you did the combat before doing Horizon, you wouldn't get an email request to meet up with Miranda afterwards. Sounds like doing the combat early and then staggering the meet ups as you do the different regular missions with the later crew (like Grunt and Miranda) and then doing just the party just before Chronos would make a lot of sense. I'll have to give that a try sometime. I don't like doing the Citadel Party after the end of the game because Shepard being so pensive about it likely being his/her last party makes the most sense just before the final battle.

On xbawks: From Ashes, Leviathan, Omega, Citadel. And I had bought an played Leviathan before beating the first time. In fact Leviathan was where I met lolbanshees for the first time (making the ardat-yakshi monastery sorta anticlimactic). However I had not yet installed the Extended Cut, so the ending was even more disappointing because of even MORE loose ends that weren't tied up.

On Origin? I paid for Javik because EA needs money, but all the other DLCs ARE TOO FIRETRUCKING EXPENSIVE AND COST MORE THAN ME3 ITSELF WTF EA.

From Ashes
The first two missions of Leviathan
Citadel clone mission
Omega
The final mission of Leviathan
The rest of Citadel

I do Omega the earliest, either as soon as I get it or not too long afterward.  It's not my favorite dlc, mainly because it separates me from my team.  It's almost like ME2 all over again in that respect.  However, I like the war assets it brings so I don't want to avoid it.

Next would be From Ashes.  This one actually used to be first until I discovered I could have Kaidan with me.  (It's always Kaidan.)  I wait until after the Cerberus coup attempt and then hit this one.  I also can then take Kaidan to Tuchanka to rescue the turian squad and disarm the bomb.

Leviathan is up next.  This one I often intersperse throughout since nothing is immediate.  I don't really need it for anything other than the war assets but I enjoy it.  I like the LI responses to Shepard's reckless behavior.  It also delves more into the lore of the series, something less true of the other dlc.  (From Ashes, I guess, but I'm not as invested in Protheans as I am the Reapers.)

Last is Citadel, which comes after Horizon.  I like the maximize the attendees of the party and the meetups with the current and former crew who just can't get enough of me.  I know, storywise, it has flaws in even existing, but I like that it's a last hurrah before going off to face certain death.  The get to spend some quality time together.  For a forced shore leave, I probably would have not done all the meetups and stuck with the party instead.  It could be done in a single night of leave, or however long space tire rotation takes.  I also didn't mind the clone stuff, though I kind of wish there had been a way to actually save it.  The archives were also great because we got into lots of history of Citadel events.

1) From Ashes, early in the game to acquire Javik
2) Omega, after the coup
3) Leviathan, progressing through out the game
4) Citadel, after Horizon

That's how I do it.

From Ashes: I do this near the start of the game. Either before or after Priority: Sur'Kesh.
Leviathan: After Priority: Thessia but before Priority: Horizon.
Omega - After Priority: The Citadel II but before Priority: Perseus Veil.
Citadel - Post-Game. The tone doesn't fit anything else.