So after you beat story and have completed all the white mage job quests, I believe you have to complete them, you will be given a quest for your relic weapon. Now the first part is easy...just open a chest in some mines....the next part however requires you to obtain a lvl 50 rod that can only be made by an alchemist. The rod is http://xivdb.com/?item/2049/Madman's-Whispering-Rod, the recipe is level 55, apparently you can make it at lvl 47 alchemy, so if you want to make your relic weapon on your own you will have to get alchemy to that level. You can also try to find someone who has alchemy that high and hope they will make it for you without you losing your house, car, kids, significant other.
On ultros I found someone selling this rod but it was going for ~170k gil...this price should go down as more people can make it but it will probably still be extremely expensive. I should have made this post when I still had access to the game so I could check out more info but hopefully this helps someone out there.
If you got any WHM questions feel free to ask and hopefully I can give you the answer you want.