I have chronic Asthma, and a rubbish peak flow, so riding hills is a real challenge. Here's what worked for me.
1. First of all, most importantly, start off by slowing right down. Forget your speed, gearing and all that.ignore what everyone else does, this is about you.
2. Concentrate on measured, steady and deeper breathing than you would normally. Practise this, most people without Asthma never consider this. Shallow breathing for Asthma sufferers here is your enemy.
3. Keep focusing on the above, and in addition now select your gearing to ensure you don't blow. I only ever use the bottom 3 gears climbing.
4. Stay seated, unless you absolutely have to get out the saddle. For asthmatics, this can cause you go blow up more easily.
5. Practise on long steady climbs where possible to begin with, to build expertise n capability.
6. Increase and build on this.
Good luck!