I also did Axton Solo. I started with the Guerilla tree and generally stregthened the turret and relied on it to kill the enemies before I mop up the balance with my guns, unless it's some kind of boss where I will gun it together. Basically, this will save my ammo since Axton does not have ammo regen. I finally reached my goal of the tier end (Double Up) where there was an extra gun and both shoot slag.
In between, I also invested in Preparation of the Survival tree. This gives some inbuilt heath regen when my shield are fully recharged. I find this useful though I have a health regen mod which will be used in emergencies, because I would usually reserve the mod slot for a sentry mod, to reduce the turret recharge time and add additional bursts.
When I reached Double Up (end of Guerilla tree), I respec and tried experimenting with Gemini (Survival tree end tier) and Nuke (Gunpowder tree end tier). Gemini is pretty weak, in my opinion. When you have a few enemies, they can focus on each of the two turrets separately, and possibly not many of stronger enemies will be killed since the total firepower is not concentrated on one enemy at a time. The Double Up is definitely better since there are two guns shooting one enemy.
Then when I tried Nuke, I've found it the best end tier there is. With Nuke, you can just throw and immediately the fiery blast (like a mega grenade) will take out quite a few of the nearby enemies. Sometimes with this, I can immediately reclaim the turret so that it will recharge sooner for the next use. It's like having and using a mega grenade for free. The turret health is also doubled. Then there is the Longbow Turret skill enroute to Nuke. Like a long range missle, when I am far away, just throw it far away where the enemies are and the blast will take out many of them and then the turret guns will mop up some while I stayed at a safe distance. Also notable in this Gunpowder tree are Impact, Duty Calls, Battlefront and Ranger which add additional damage to my guns. After reaching Nuke, new skills are added to Sentry (to boost turret firepower) and then to Laser Sight (it really helps to reduce misses turret misses). Adding to Preparation in between reaching any tree end tier is a must to regen my health.
Start with Guerilla tree and accumulate enough skills to reach Double Up. Us ethe turret to distract and kill for me and save my ammo.
In between, spare some skills for Preparation (and strategically run away to regen health) to reduce dying.
Upon reaching Double Up, try Nuke which I find to be more devastating and versatile (long range mega grenade!)
Then add new skills to Sentry and then Laser Sight to further boost the firepower of the turret. I am now at Level 40 and into the DLC so that's all the points I have at the moment.
Hope this helps.