Are there any major quest conflicts that will arise from this? I have completed Morag Tong once, but never got very far in House Hlaalu. I know that the grandmaster writs have you assassinate very high priority targets, and those conflict with House Telvanni for sure, but what about house Hlaalu?

My idea for the current character I have (Imperial) is a ruthless Morag Tong assassin, who uses House Hlaalu as a cover story for his true intentions, and Morag Tong assassin as a cover story for his loose affiliation for the Blades (which he is more or less svcked into). I really like the idea of an undercover, undercover stealth character, and I feel very immersed with the fact that nobody would question this Imperial outlander, due to the fact that his shadiness (working as a spy for the Blades) is just because he is working with House Hlaalu, or, if digging deeper, the Morag Tong.

But enough about roleplay: What about quest conflicts? Should I do Hlaalu first and Morag Tong last? Currently I am just in the Morag Tong and just got the mission to Vivec from Caius. Aside from that I have been freelancing, clearing caves, and killing 6th house cultists that I find in cities by provoking them with ash statues.

How should I proceed as to break as little quests as possible?

I don't think there's any conflicts, until you get to the final Grandmaster Writs (which require you have finished the main quest of the game) and then you have to kill a Hlaalu council member, Dram Bero.
He is needed for the Hlaalu main quest, and the main game quest.

How about he finds out about the Tong on this Vivec quest, begins to love the idea of becoming a member, and only after that inventing the "Hlaalu cover"? After all, he wouldn't have any other intentions to join Hlaalu except for the cover agenda, now would he? He could of course join Hlaalu first, but he should at least know about the Tong and appreciate them beforehand.

I checked a few guides, and there should be no conflicting quests between Hlaalu and Morag Tong. Although I do recommend to leave those Grandmaster writs alone, just in case.

This double cover agent thing sounds real good! Once the deep diggers find out about the Tong, that should appear motive enough for the cover and the digging should stop there. Of how wrong they would be... One only wonders how many IRL politicians and stuff have this kind of triple role, if not more... :I