Phase one is pretty boring ones you get the details right. Let the Paladin's worry about cleansing and keep the HoT's on the Tanks, depending on your group setup you might need to take care of Infest. If you do than make sure the Raid is ready to take it by throwing Rejuvenation all over the place. Our LK team has a DiscPriest taking care of the Infest which, in my opinion, is a bit better.
If all the adds and Necrotic Plague are being handles as they should than the first phase should be easy enough to get past.
You should be able to get into the first transition phase without any horror's alive, a few ghouls might be still up but should go down fast with whatever remains from the Necrotic Plague or dps should burn them down easy enough.
The Lich King starts casting Remorseless Winter, which is a big AoE attack as well as Pain and Suffering, which is a DoT thats being put in random raid members. P&S also deal damage to nearby targets to spreading out in three groups helps you counter this part a bit. As a RestoDruid you want to keep Rejuvenation running on as many people as possible, Wild Growth should also be used on cooldown because of people stacking in groups. Raging Spirits will start spawning not long after that start of the transition phase, they have some aggro on the person that spawns them, Tanks should pick up as soon as they are active but keep Swiftmend ready just in case. You can cast Regrowth on people that will spawn a Raging Spirit to counter any incoming damage and use the long HoT to help on the AoE raid damage that comes from Remorseless Winter.
When starting phase two everyone should run from the side to the middle of the platform, which is where the LK will be tanked (center or slightly off center). Both Tanks will get damage as soon as the LK is picked up and placed into position so keep Rejuv & Regrowth running on Tanks, I use Lifebloom as a safety measure when all the other healers where running to the center so the Tanks received that extra bit of healing, also the 'Bloom' is useful to counter damage while the raid gets into position.
Again like in phase one you might need to take care of infest, if that's not assigned to you than keep in mind that the person doing so might get picked up by a Val'kyr and won't be able to perform his/her job. Work as a team!!
Keep and eye out for Soul Reaper, just like on the Infest part if the main Tank healer is pick up you might need to fill in the needed healing on the Tanks. A few Nourish crits do well on this, and its even better if you have Living Seed in your talent build.
Ones you got phase two under control it also become rather easy. Although a bit of bad luck on the Defile and you're perfect run can turn into a nightmare!
After the second phase the LK goes into an other transition, this one is however a bit harder than the first. Healing wise nothing changes compared to the first transition phase, there will be more Raging Spirits to take care of so those might cause additional damage. Something if have found out however is that at the start of the second transition phase the raid takes a big chunk of damage from Remorseless Winter, pre-Hotting the raid is a very nice way to counter this!
The LK keep Soul Reaper and Defile, but stops doing Infest. Again this phase is all about control, if your raid can kill the Vile Spirits before they reach anyone than raid damage should be kept to a minimum. If not than you're in for a whole lot of fast reaction Nourish/Swiftmend action. You probably won't have the power to kill all Vile Spirits before they start flooting towards the raid, so keep a very focussed look onto your healing UI to counter damage as soon as it hits, there will be time that people will have two Vile Spirits on them, make sure you only use Rejuv+Swiftmend when its really needed. Rejuvenation doesn't add much to this phase but it does not mean its useless, keep the Tanks healed with HoT's. keep Wild Growth ready to heal the raid+give a nice boost to Nourish. If you feel you have the time to keep Rejuv running on the raid than by all means do so. it can only benefit your Nourish and Swiftmend can be use faster this way. There will be a lot of spike damage in phase three, using Regrowth on the raid is also a good idea because of the long HoT to keep Swiftmend going.
In the third phase the LK also gains an ability called Harvest Soul, it does a lot of damage per second and if its target survives the damage he'll be pulling into Frostmourne. Once you're in there you have to heal up Teneras as he fights the Warden. Here is a very nice vid on what to do when: click
Once you get him down to 10% the last phase begins, see for yourself what happens! :)
I hope this post can help you in any way, if I've missed something or said something wrong please let me know!