For this achievement, you'll need to find the designs for two Mithril items and forge them.
The design for the Fire Bow is found in the eleventh story chapter 'Helm's Deep' after you topple all of the Urak-Hai ladders, behind some Mithril Brick that must be destroyed by Berserker or the Mithril Fireworks.
The design for the Ice Bow is found in the fourteenth story chapter 'Cirith Ungol' near the end of the level, when you're searching for orc armor for Frodo and Sam.
Both bows cost 8 Mithril Bricks each to craft/forge, so this achievement will set you back 16 Mithril Bricks. Once you forge both Mithril items at the Blacksmith in Bree, the achievement will unlock.