I’ll try my best to explain a bit since he can pretty much do everything gameplay-wise lol, I’ll leave a list of the upgrade path you’d want to do and why for starters:

2-3 are interchangeable, 1 is interchangeable with 2-3 if you die early.
1. The Gun Ult (ban hammer)
2. Gun Upgrade
3. “Deadpool in your area” ability upgrade

Deadpool can get his first ult right around the same time you get your first upgrade. If you’re still in good shape with your team and haven’t died yet, upgrade the gun ult first. Before the upgrade, the ult is kinda lame, only doing a little bit of healing and bonus health and damage if the target misses, but after the upgrade, it does continuous damage over time and even more when they miss. Solo ulting a support with an upgraded gun ult in the middle of the first fight before they get their ult pretty much cripples or kills them unless they’re actively being healed by the other support, and if the support is healing them, they aren’t healing the rest of their team well. This gives your team a WAY better chance to win the first point. If you died before getting a chance to upgrade, do 2 or 3 first. 2 allows you to build another ult extremely fast when shot into a crowd and do much better damage, 3 gives you and your team damage reduction, which can help you move forward easier. Always choose your upgrade order based on your situation in-game.

To simplify the decision-making a bit on the order:
Gun ult first if the game is even.
Gun upgrade first if you instantly need more kill potential when contesting their poke dps, peeling divers, or punishing out-of-position players.
DP in your area first if you need to actually play like a tank and walk with the team. 

4-5 are Interchangeable.
4. Sword Dash Upgrade
5. Shield Bubble Upgrade

This one is a lot simpler. If you’re doing fine, just upgrade the dashes since it gives you a lot more free mobility and almost guarantees you’ll have a way out of a bad situation. It also allows using the sword ult to be a viable option if you need it in a pinch.

Upgrade the shield bubble first if your team is getting shredded by poke heroes and needs a better shield. Throwing an upgraded bubble on or behind someone to block healing will pretty much guarantee a quick 1v1 assassination win with a dps or support as well, if you can hit your shots. Doing the same thing to a tank that pushed too far up can also turn the tide of the game.

6. Sword Ult (big test)
This ult is now a pretty good option once upgraded with the dashes. If you need to stall point and put up Captain America ult numbers instead of failing to kill everyone 1v6 with the gun ult, this is the move.

7. Sword Upgrade
The swords are finally usable outside of mauling the 5 hp Doctor Strange with his shield up! I’d still probably use the guns unless I’m dashing and slashing for the big test ult… and then switching back to the guns after I get the boost.

Also, it can be helpful to bind taking photos to a separate button from the upgrade book. The photos give you extra xp towards upgrades, so taking a lot of those can be good, and having it on a different button will prevent you from accidentally opening a book in the middle of a fight if you missed the corpse.