Healers in this game pretty much have to be magicka. So your sets and weapons must keep this in mind.

And Dragonknight Healer must use a restoration staff on at least one bar. The only class heal we have is Cauterize, which 1. only goes off once every 5 seconds unless you spam it, and 2. will not heal you, only those around you. So you cannot make a healer without using the restoration staff abilities. (I would recommend them even for a Templar healer, of course! But for a Dragonknight, it's a must.)

All of the spells on the restoration staff are useful, so it's more a personal preference there, although in my opinion Steadfast Ward --> Healing Ward is a must as it's your only real burst heal, and you need one of those. I can go into more details on these if you want, but there is a lot out there for that.

Your other bar is up to you. My healer likes using a destruction fire staff for DPS and debuffs, plus Elemental Drain to help restore magicka to the group.

You want to put Igneous Shield on your bar and spam it for the 6 seconds of major mending.
I like Molten Armaments as well to buff the party. These are both Dragonknight skills, and casting either also gives the group minor brutality if you put a point in that passive. Healing is also about buffs and group support, and shields in the case of Dragonknights.

Armor has a lot to do with preference. Mine is currently running with Trinimac's Valor and Julianos with one piece Molag Kena, but there are lots of options. Spell Power Cure is on my wishlist, as is a CP160 Restoration Staff for the stamina boost to those I heal. (Aside from that staff, only Templars can give stamina to other players). You do want to make sure you have decent magicka regeneration as you can't always predict when you need to heal. For me, 2k is the magic number.

For CP, I only have a few into Blessed, but a lot in Bastion and also the magicka regeneration and reduced costs.

Max magicka, spell power, and spell crit are all useful stats for a healer as well, so I have been boosting those to boost my damage.

This build on Deltia also has some good pointers. It's outdated, so I would not recommend copying it, but it does a good job explaining the reasons behind things and can give some nice food for thought:
http://deltiasgaming.com/2015/06/01/dragonknight-healing-build-the-infernal-warden-reborn/

I've been playing a DK healer for about a year and a half now. Hopefully some of this is beneficial for you

Right now (and for the foreseeable future) the BiS healer set is Spell Power Cure. It pairs very nicely with Julianos, Willpower, and Torug's Pact. I recently gave up running dual swords on my off-bar (both of which were Torug's Pact for the 2set bonus) to go back to a Master's Restoration staff, however another, arguably better, option is to run a destro in order to maintain Elemental Drain for the group. If you use a resto on your back back instead, make good use of Siphon Spirit.

In August, it appears that the best set to pair with Spell Power Cure will be the Gossamer Set. At that time, myself and both of my core trials healers will be running SPC/Gossamer. After a pretty length discussion and a bit of speculation, we think Gossamer will outpace Twilight Remedy, Infalliable Aether (which we are going to have someone else run instead of the healer), and Worm Cult. We'll make up difference in magicka sustain through Siphon Spirit, Necrotic Orb, and Ele Drain.

Up until TG was released, the setup for a DK healer was much different than it is now. It used to be more effective to use the Ritual Mundus (all divines gear), paired with a heavy investment into Blessed, Elfborn and some into Bastion. However, once TG released, DK healers fell onto the same tracks as virtually every other meta magicka build, with the vast majority of our focus being on max magicka and spell damage. Certain changes to CP and the addition of new sets, changes to Major/Minor buffs, etc. has made it possible for us to make up the difference in pure healing potential we would have gotten from Ritual and Blessed with just stacking magicka and spell damage as well. As such, our CP allocation changed to a more damage-oriented build in order to contribute to dps checks and speed up certain content.

Here is how I recommend you set up your bars for a trials setting, noting that the back bar is flexible (and the most likely to change for 4-man content):
Bar 1: Mutagen, Igneous Shield, Healing Springs, Healing Ward (flex), Inner Light, Shooting Star
Bar 2: Combat Prayer, Siphon Spirit / Ele Drain, Healing Springs, Cauterize, Inner light, Aggressive Horn

The priorities in your rotation are as follows:
1. Combat Prayer - at all times
2. Spell Power Cure - at all times
3. Siphon Spirit / Ele Drain - at all times during single-target fights (and whatever else you can conveniently put it on)
4. Major Mending (via Igneous Shield or heavy resto attacks) should be active for the more intensive mechanics, but can be situationally foregone with the right gear and the right stats.

Your Warrior CP follows pretty close to the magicka meta, with Hardy/Elemental Defender/Thick Skinned taking the majority of your points in a 2:1:1 split, respectively. At least 10 points into Quick Recovery, and however many you feel comfortable running into Bastion (yes, it's worth it for your Igneous Shields and Healing Wards).

Mage CP is contingent upon how you want to play. If you want to contribute damage, then the vast majority of your points should go into Elemental Expert and Thaumaturge. In this case, I recommend doing something like 70pts. into each, 15 into Elfborn, 10 into Blessed, 1 into Spell Erosion, and 1 Staff Expert. If you care not for contributing damage or don't need to, then you can take a substantial number of those points from Elemental Expert and Thaumaturge and distribute them between Blessed and Elfborn for more potent heals.

As for Thief points, it's really up to you. I encourage you to most heavily weight Magician, for reduced cost, and then put whatever you have left into Arcanist for recovery.

Note that having a large number of points into Blessed and Elfborn will substantially improve the potency of your heals, however, it is largely unnecessary. Prior to TG, I was in a fully dedicated healer build, with no points in Elemental Expert and no points into Thaumaturge. Almost everything was dedicated to Blessed and Elfborn, with a handful in everything else. Fully buffed in 4-man dungeons, I was seeing 10k crits on my healing springs on a single target. This is way beyond what is necessary, even for harder trials. As such, you really can afford to supplement your damage. Your heals will still be strong enough to compete.

Attribute allocation is standard; aim to sink most or all of your points into max magicka. Aim to have 40k+ magicka, 17k+ health, and 2.5k+ unbuffed spell damage. You may want to use the Thief to supplement your crit, but you can also use the Mage or Apprentice. Your mundus largely depends on your playstyle and preferred content.

If you aren't getting Major Sorcery from your tank(s) or other DKs, then use potions. Siphon & Ele Drain, along with Springs, Cauterize, Inner Light, and Combat Prayer are all far too valuable to give up. You could trade Healing Ward for Igneous Weapons if you like, but remember that Healing Ward is a better heal than Cauterize. It not only provides a stronger benefit (the shield) and a stronger heal, but it's also got a longer range on it and can heal the caster, which Cauterize cannot do. Cauterize is best utilized on the off-bar as a buff you simply "keep up" for its periodic benefit, or on a dps bar, utilized as a clutch heal without having to bar-swap. In trials, someone should always be running Igneous Weapons for the group, which is quite easy if you have a DK tank (and is usually preferred by tanks, since it gives back some stamina upon cast).

Also note that recovery is not all that important for a DK healer. As long you actively manage your light/heavy attacks for the ultimate gain, and are constantly healing, you can almost solely rely on ultimates for sustain. A good target base for magicka recovery is around 1k for a DK healer.

The "obvious" choices for race would be your magicka races. My DK has been an Argonian since I first created him at Early Access (back then he wasn't going to be a healer, but things change and I like doing unorthodox builds). This was way before DKs were "good" healers, and I've been doing it successfully since. In August, Argonians are becoming much more powerful as healers. But if you take anything away from this segment, it's that you can do any end-game competitive content with a DK healer, regardless of racial selection. For example, when vDSA was the highest progression content, most of my successful runs (including my very first clear + leaderboard spot) were on a DK healer - and at the time DK healers were literally unheard of. I had people shooting down the idea left and right, but it didn't matter because I proved it could work. After that, people started taking my DK in vDSA on a regular basis, because not only could he crank out some serious heals, but he also had a wide variety of class/armor skills at his disposal to heal/tank bosses like Hiath for 100% burn methods, etc.

DK healers are extremely powerful in a stack. The same argument can be made for Templars very easily as well, but if anything they are paralleled. Where Templars tend to outpace DKs is in two areas: 1. the ability to restore the group's resources, and 2. when everyone is spread out (like way spread out, to the point where you can't hit more than 2-3 people with springs at once. DKs, on the other hand, excel at providing powerful damage shields for their group, along with buffs, such as Minor Brutality, and if you choose Major Sorcery and Brutality as well. One of the more disappointing downsides (imo) for DK healers is that they lack any useful synergies for the group. Take Twilight Remedy, for example: the group damage shield component for Magma Shell used to be synergy-based, which would have been fantastic in combination with Twilight Remedy, except it was changed in the exact same patch that the set was released.