This is exactly how to do the quest, from start to finish...
While fighting [npc=52558] you loot the [item=70997]s from [object=209033]s which randomly spawn around the edge of the area (note: possible link to the volcano spawns, but those are random as well). Gather three of these and create a [item=70996], which is then used just before he uses [spell=97282]. He has to stand on it almost directly so make sure you're pretty close before using it. Loot the [item=71008] afterwards.
While fighting [npc=52498] you loot the [item=70999]s from [object=209037], three of which spawn when a [npc=52581] is killed. Gather them and create a [item=70998], which is then used on top of the web just before she uses [spell=99052]. Go back up once it's safe and loot your [item=71017].
While fighting [npc=53691] you loot the [item=71000]s from [object=209035]s, one of which spawns when a [npc=53713] is destroyed. Gather three of them and create a [item=71001], which is then used just before he uses [spell=100002]. The spear has to hit the focus so make sure you're close. Once the fire has died down loot the [item=71015].
While fighting [npc=52530] you loot the [item=70994]s from [object=209036]s, one of which spawns when a [npc=53898] egg hatches. Gather three and create a [item=70995], which is then used just before she reaches [spell=99925] (100 energy). It doesn't matter how close you are so just drop it where you're standing at around 90 energy and then loot your [item=71016]. Note that only two of the fragments spawn at a time, and that if you are usually flying around you will have to switch roles (or just have no-one flying at the start) to do this.
A very important note if you want to get Dragonwrath as soon as possible: do NOT kill any of these bosses. You can loot your charged focus then just wipe and move on to the next boss. Also note that if your raid is willing to let you do this, it only takes a few minutes to do the quests in between this one and Time Grows Short as there is no fighting or anything involved, it's just travelling and cutscenes. To make this as quick as possible I'd advise setting your hearthstone to somewhere near Amber Ledge, like [npc=25278] for Horde or [npc=26596] for Alliance.
Once you have all four of the charged foci go to the portal near Beth'tilac and use them. After entering the portal you begin the encounter with Volcanus, which is a three phase fight.
Phase 1 - just interrupt and kill all of the [npc=53793]. You can burn mana and cooldowns to make this easier as phase two starts only when the player on the quest tries to loot the branch.
Phase 2 - [npc=53825] will need to be tanked at the entrance portal. He casts entangling roots which does damage, but only dispel them AFTER he does his stomp. If you dispel them beforehand you will get knocked back an enormous distance and die. Once he does his stomp dispel them as soon as possible. Small treants will spawn periodically and act identically do the transition phase adds on Ragnaros: they must be killed before they reach the treant. Ranged should deal with them. At the end of this phase he becomes friendly and casts [spell=100274] on the player on the quest.
Phase 3 - [npc=53833] will fixate on the player on the quest (who is transformed into a huge tree of life) so tanking will not be an issue. Use ALL DPS cooldowns and burn him down quickly. There are a few spells to avoid, so as always do NOT stand in the fire. Healers have to compete with constant raid damage.
After that just loot the branch. Simple!