As mentioned, Falconer is mainly a DPS character, but can still offer some utility to the team in the form of 2 abilities in her kit.
Primary Ability: Shock Drone (5 Charges, 10-second CD per charge)
One of the more complicated abilities in DRG Rogue Core. You send out your Shock Drone to a target location, after which it will scan and search for enemies. In case it doesn't find any, it will return to you. If it does find an enemy, it will shoot and electrocute them, consuming one charge. Its course can be changed mid-flight and can also be forced to target a specific enemy.
Secondary Ability: Thunder Rod (120-second CD)
Place a defensive zone down for 15 seconds that gives both you and your allies +100% electric damage. Great ability for enabling elemental DPS.
Falconer sends out her drone to a target ally to revive them. One of the most powerful abilities in the entire game.
Bio Booster Upgrades
Falconer has two active starting decks. Mixed Tactics, which is more centered around her abilities, and Storm, which is fully focused on DPS.
Both decks are stacked with solid cards, but which to pick depends on what you want to accomplish. If you want to take a more supportive approach, take cards from Mixed Tactics, in the DPS case, Storm.
Workbench Upgrades
Falconer workbench upgrades can be very solid, but they do depend on your build a bit since they all focus on her drone.
If you want to play around with elemental damage a lot and improve the drone, then take more charges or more projectiles, since it will massively improve the DPS.
What Upgrades to Prioritize
Every time you get enough experience to upgrade, an important choice awaits. Here are some of the types of upgrades you should prioritize.
You mostly want to build out Falconer as a full DPS, and while you can build CDR for her abilities as well, I won't go in-depth on that since the build stays the same besides the CDR upgrades.
Anything with Gun Damage. This includes Gun Damage, Elemental Damage, Crits, etc...
Movement Speed is always very nice when trying to move around as the Falconer.
Weakpoint damage is generally solid since you will be looking to focus on firing guns as Falconer, so this will never hurt.
If these choices get taken or don't appear, take HP or Armor upgrades.
Artifacts
Falconer gets massively improved by some upgrades, which can make her DPS insane at later points in the run.
Glass Cannon. You gain 150% extra damage but lose 90% max HP. It's the best damage Artifact for her gun builds.
Frenzy. Every enemy you kill gives you 20% extra damage dealt and +5% damage received for 6 seconds, this effect can stack. This gives you a ton of extra damage, and the downside is not really there.
Ceramic Defense. Gain 40% damage reduction but lose 1% for each 1% of missing HP. Very good if you have Glass Cannon, otherwise it's not that good.
Rampage. +1% Damage per kill. When getting hit, reduce the bonus by 5%. Another risk-reward one with almost no downside if you can position well.
Tarred Hands. 50% increased reload speed. Very simple but still very strong.
Best Weapons
Weapons for the Falconer all center around her DPS playstyle, meaning you would preferably have carry-oriented guns.
For the gun playstyle, you want weapons that, when upgraded, can single-handedly melt bosses.
By far the best gun when upgraded is the Bolt-Tec Decker SMG. When upgraded, especially the magazine, which is its only weakness, this weapon is a monster and should be nerfed with how good it is.
Bolt-Tec Silverback HAR-50. A very solid assault rifle that only gets stronger with upgrades, especially if you get the chance to give it a bigger magazine.
The Deepcore GK2 is a weapon that was already very good in DRG, but with the ability to upgrade it now, it's on a different level.
Lead Storm Powered Minigun is another gun that's solid at a base, but when upgraded is on a different level, especially when it has armor penetration.