Basically the way you did it, Accept the quest on a horde alt and complete it on your Alliance character.
Now where this went foggy for you, was the lack of follow up quest, Unfortunately you will only get the follow up on your horde character, So you would have to log over and accept the next part on your horde character then swap over once again. Rinse and repeat until all are down.
i did this actually. after i did the quest i switched over to the horde alt to accept the other an it wasnt there. it wasnt giving me any other quests....i logged out, deleted my cache an still nothing so decided to try later. when i came back later i had all of them completed on the achievment...but only to my horde alt...ugh
i was looking threw the forums though an someone made a similar topic. Someone mentioned that you have to be a specific lvl to battle those in certain zones...my little horde is lvl 12 an i was only able to take the first quest.....someone recommended making a dk an going back....ugh but i dont want to waste more time if this doesnt work lol
what lvl alts did yall have to make in order to complete the tamer achievments? clearly you need a horde/alliance alts/mains but do they really have to be specific lvls to take all the quests? I read you only had to be lvl 5 to take the quests an then could just do them on ur mains -.- so im not sure what to do now