The First Battleborn: Get a first blood as Ghalt
This one is pretty self explanatory. You need to get the first player kill of a match. Ghalt isn't a slow character so being on the front lines of the fight shouldn't be too much of a problem. Save your hook for when a weak enemy tries to escape and don't be afraid to steal kills. This should just come naturally with playing Ghalt because of his good damage output and skills that prevent escape.
Social Engineering: Kill 100 Jennerit enemies with scraptraps
This should 100% be done in the campaign mode due to the only Jennerit enemies in the mulitplayer being other players and the thralls, of which there are few. Even in campaign this will take quite some time due to the scraptraps cooldown. I personally suggest doing either The Heliophage or The Renegade missions, both of which have a lot of Jennerit enemies. When fighting Jennerit, weaken them pretty significantly then throw down a trap in front of them and either lure them in or pull them with the hook (beware not to kill them since the hook does a small amount of damage). As for your Helix upgrades, I used the right upgrades on levels 1,2,6, and 8. When you unlock the mutation for level 8, use that instead since it will get you your scraptrap back quicker.
Ghalt's Call: Pull 25 enemy Battleborn with the Hook
Pretty self explanatory. Just use Ghalt's hook ability on enemy player to pull them in. You should get this in 2-4 matches of normal play.
The First Shots of the Solus War: Deal 500,000 damage with Ghalt's Revolver Shotgun
Once again, self explanatory. The revolver shotgun is Ghalt's main weapon and the only consistent source of damage he has, so just play normally and rack up the damage. I'm not sure because I haven't tested it, but I believe his ultimate would also count towards this, since he pulls out a second shotgun revolver and dual wields.
First Round Draft Picks: Complete 3 matches with Deande, Melka, or Kleese on your team.
This is kind of luck based since you have no control over who your teammates choose (unless you're playing with friends). Deande and Kleese are both fairly common characters to see so you shouldn't have too much trouble. If this ends up being a pain then you can try and set up a boosting session either here on TA or through the LFG feature on Xbox One.