Can't Touch This

STOP ! HAMMER TIME

Burning Cinders is the dot that you get from touching the fire lines coming from the first phase spear.
Storm Cloud is the debuff you get from the lightning lines coming from the second phase spear.
Rushing Winds are the tornadoes in the second phase. They behave like the tornadoes on Blade-Lord Tayak.
Frozen Blood is the dot/slow you get from touching the ice lines coming from the third phase spear.
Frozen Solid! is from getting stunned by a frozen cyclone in the last phase, which is incredibly difficult to do unless your tank is holding the boss in an idiotic place.

the most difficult criteria to meet is probably the Burning Cinders. if your stack group gets separated by fire lines, it might be necessary for someone to cross over to meet up with the others. note that the range to trigger the dog's fire is large enough that if your group is only split by one fire line, you can straddle the line and still be in range of each other. you could also ask the others to always meet up with you for one kill since this is a personal achievement.

i can confirm that, like Do a Barrel Roll! and Achieve-a-tron, it is not required to meet all of the criteria in the same fight.

edit: one of the people in my raid group this week died mid-fight and got the achievement by default. this is opposite the behavior of Do a Barrel Roll!, which requires you to be alive at the end. like JeremyAvalon mentioned below, this might be an oversight.

Soloable in 7.1

I simply burned him down in each phase before he could do any attacks.