You can find the Ring at Tear's Point (you can also get it in Chocobo World or with the "Search"-Ability of Rinoa's dog - good luck if you try...). Then you need 6 Malboro Tentacles, 6 Steel Pipes and 6 Remedy +. When you've got all of that you can use the Ring in the inventory to get Doomtrain.

How to get the items:

Malboro Tentacles: Fight/mug Malboros oder change Malboro-Cards into Tentacles
Steel Pipes: Mug Wendigos
Remedy +: Buy 60 Remdies and change them into Remedy + with Med LV Up