The impact of running can often wind people who aren't used to it. I took a break from running while gaining most of my weight and on return got my first taste of this.
The first and most mentally painful thing I had to do was slow down, a lot. I am a natural runner so hated this with passion. My technique and style had to be adjusted to suit someone 1/3 heavier than my distance running weight and this meant taking it steady.
Intervals are great for increasing lung capacity and getting you used to breathing during continuous impact exactly as PLB says. However I found it was the movement of running that affected this so walking on the recoveries where breathing is hardest, was no good for me. I slowed to jogging at a pace roughly the same as walking but maintaining the string of small jumps that marks the difference between jogging and walking. Initial was minute run, minute and a half jog, but I was used to distance, set your distances or times to suit you.
Look seriously at your technique and running style. Ideally you want to be breathing freely and moving relaxed and light. If you feel every landing in your jaw, you're hitting the ground far too hard and likely from too high up. Your feet should skim as close to the ground as possible, knees up running is not good form. Your shoulders should be relaxed, tension here will put strain on your breathing.
Finally there is the old classic of getting decent running tops. There are a number of them that provide too little support meaning you bounce too much affecting your breathing, or too much literally restricting your expanding ribcage. I won't name brands because everyone is different, and good for one is not good for all. I can sort of empathise with this one, I went from having mostly visible ribs at my distance running weight to actually having pectoral muscles visible through clothing, and no-one tells you that muscle can bounce about when running too, you are left to figure this out for yourself, and men don't get sports bras, there is no equality.