Evil Dragon Godesses Not Welcome
Due to the most recent Neverwinter update this has become MUCH easier to obtain. You cannot solo this (which that would be impossible if it was possible). Once its a new hour you can search for a group to do Tiamat. The game will search for everyone for you and you will be thrown into Tiamats temple once everyone has readied up.
Thanks to the new update there is only 2 tanks for it and basically all DPS characters (hopefully you are a DPS character).
When things get going everyone must kill the summoners based on color, there is 5, just like 5 heads for Tiamat. If you beat a summoner you can take the orb off the ground and equip it to your potion shortcut, this will give you bonus damage towards the head.
Once all 5 Summoners are dead you must Kill Severin who is in the middle of the entire area.
After killing Severin 3 Clerics will appear and you must defend them from oncoming enemies until the blue bars are full.
Now once all the blue bars are full everyone goes from the first head on the left heading right, weaken the heads down below 1/4th health and go to the next head. If all goes well you will repeat defending the clerics 3-4 times (anymore and you will likely not beat Tiamat).
With the newest update Tiamat is nothing compared to what he used to be considering the amount of tanks have decreased and there is more damage being dealt.
Here is a video demonstration by Zach Attack Gaming:
If you are looking for a good character build refer to mmominds.com, make sure you are looking at the most up-to-date mod (which I believe right now is mod 9). If you already have a high level character you can grab Retraining tokens from the Zen Market, nothing is perminant but if you want something reset it will cost you (can use in-game astral diamonds to convert to Zen).
Any other questions please ask in the comments.