You can use the Rocket Jumper to practice rocket jumping without dealing self-damage, however, it also does not deal damage to enemy players at all, making it next to useless for the Soldier Gunmettle Contract Advanced/Bonus Objectives.
I wouldn't worry too much about self-damage. The Soldier has the second-highest Health (after the heavy) at 200HP, so he can take a bit of self-punishment. It also helps to know where health-packs spawn on the map, (and potentially to team up with a Quick-Fix wielding Medic)
I generally prefer other classes, however it only took about 30 mins of practice to get the hang of Rocket Jumping for me. I followed the tutorial here when practicing:
As for completing the contract, here's what I suggest:
Maps: Capture-point with open-air points, or Payload maps. The new CP_Powerhouse's middle point is great for Soldier-spam.
- Primary: Liberty Launcher.
The increased clip size and blast damage makes clearing objectives/capture points from midair a lot simpler. You'll get a lot more airborne kills clearing these out, especially as the enemy takes damage from your teammates' fire as well, which makes up for the -25% damage.
- Secondary: B.A.S.E. Jumper
The Advanced part of the Soldier's Contract requires rocket jump kills, and more time spent in the air = more chances at scoring the kills. Paired with the Liberty Launcher, you should have no problem scoring airtime-kills. (I'm usually more of a fan of the Buff Banner for the Team support aspect, but you can't go past the added air-time this will give you)
- Tertiary: Market Gardner or Equalizer
The Market Gardener deals crits whilst Rocket Jumping, Whereas the Equaliser deals more damage, the more you are injured. The Equaliser is better all-around, but the Market Gardener is also viable.