The forge is unlocked once you reach level 1 and is only accessible in the Inn.
Once you go to the Forge you will see that you have the option to fuse weapons to a one tier higher better weapon. This achievement does not require you to make 25 higher tier weapons. It requires you to use 25 lesser weapons to make a better one.
For example: You can cycle through all your characters and pick 5 white weapons to put into the Forge. Then hold down
to fuse them to make a green weapon. (5 greens to make a blue and etc). This will count as five towards the achievement
This requires no salvage (the rock icons at the bottom) to make the weapon. You can put less than five weapons to fuse to make a better one if you have the salvage for it. You can put 3 white ones in and use salvage to fill the remaining two slots for the weapons you don't have. This will count as 3 towards the achievement.
So in summary: This is not to make 25 higher weapons. This is using 25 lesser tier weapons to make the higher one. This can be achieved through the combination of just pure 25 to make the higher weapon or a couple weapons with salvage to make the higher one. Every weapon you use to fuse will count as a tick towards this achievement.
Hope this helps understand the poor wording of this achievement. If you down vote please leave a comment why.