To get a weapon to 6 stars it must be capable of that upgrade, ie be purple rarity and have the 6 total stars at the top of the weapon portrait. The first weapon you get that you can do this with is near the end of the first area, "Yarlanda's Long Sword" you get from beating Yarlanda's first mausoleum. To upgrade the weapon all the way, you'll need basically 1 of each of the weapon upgrade items which you'll get from chests and drops. You should have found the last piece you need, the diamond, by the time you clear the 2nd area. Once you have both a 6* capable weapon and enough mats, just go to the Goblin Market blacksmith and upgrade the weapon to its maximum.
Happy Gunting