There are a handful of soulbound weapons in the game, but the first one I found (and likely the first one most people will) is called Gyrd Háewanes Sténes. It is a magic scepter that you receive as a gift upon completion of the "A Call to Court" quest: this will start automatically at some point in the second chapter of the game, after you've freed Caed Nua and gotten to Defiance Bay. You'll get a note that a messenger is waiting for you at you at Caed Nua. Go there, talk to the messenger, follow his instructions to talk to someone else at a specific location in Defiance Bay, and after that second conversation, the quest is over and you'll get the scepter.
The scepter is only usable by a wizard, priest, or druid (other soulbound weapons have different class restrictions). Equip it and make it the primary weapon (I gave it to Aloth, but choose any wizard, priest, or druid in your party, including yourself if you are one of those classes), view the details of the weapon and hit the button to bind it to the character wielding it.
After that just play the game normally. As the weapon does damage, it will gradually level up. Each level adds bonuses to the weapon. The first level requires you to do a certain amount of cumulative damage. The second level, either (a) additional cumulative damage, or (b) a lesser amount of damage to stunned opponents (for me, the first of these occurred much more quickly). The third level requires the weapon to successfully create an effect a certain number of times (the probability of the effect is about 10% per hit, so not too bad at all). I got all of these through normal game play without any special effort or time put into it.
After the third level, the weapon is maxed out and the achievement will unlock.
Note: if you want to switch the weapon to another character, you need to unbind it and then rebind it to the new character, which causes its leveling to revert to zero and you have to start over. You're best off choosing someone who you plan on retaining in your party for the long haul so as to not waste the leveling of the weapon.