These quests are very easy (minimum level 75) and you could get them done with a ground mount, but are really meant for characters with flying mounts in Northrend as they dramatically decrease the time to get them done.
Step1 :
Visit the orphan matron in Dalaran near the Alliance side's bank and get your orphan. There are two orphan's to choose from. A wolvar and an oracle (really ugly murloc).
Step 2:
Get three quests from your orphan. "Playmates!" is the only quest in this step that is different for the two orphans. The wolvar orphan has you go to the wolvar village in Dragonblight and the oracle orphan has you go to the Winterfin Retreat in Borean Tundra.
Step 3:
Both orphans want to meet the Dragon Queen in Wyrmest Temple (with different things to say to her), but they want to go to two different places for the second quest. The oracle orphan wants to see a Great One in Sholazar Basin while the wolvar wants to meet Hemet Nesingwary in Sholazar.
Step 4:
This is the easiest step. Go back to Dalaran to buy a small paper zepplin at The Wonderworks in the tradeskill section and play catch with your orphan.
After you've done that, go back to the matron to finish the quest. A minute or two later and you will get some mail with your new pet.
I wasn't really paying attention to how long it was taking to get these Northrend Orphan Week quests done as I was doing other things, but I would say 30 minutes is an accurate guess. If you like to collect Orphan Week pets like me, both pets are a must. Plus, the wolvar pet is pretty cool. :)
Side Note: I have to give kudos to Blizzard for the change to the Quest Log screen. The vertical split lets you see more quests in your list and more quest text on your screen at once. Very good change.
The payoffs:





