I find that when I run, what holds me back is not my legs but my lungs. Is there any way to improve breathing/increase lung capacity?
You have got it all wrong. It is not your lungs or breathing capacity that is the problem. The actual cause is your oxygen delivery system is very poor compared to your lower body endurance capability. For most inactive people it is the other way round. Their legs start burning first before they run out of breathe.
When you run, your heart rate goes up and when you reach around 80% of HRM you will start breathing heavily or maybe earlier. It is your heart trying to get more oxygen for your body to use as well as elimination of the carbon dioxide that trigger a faster breathing pattern. You have to train your heart to be more efficient for more oxygen delivery per stroke. A slower beating heart will result in less stress when breathing and the main concentration of the lungs is directed to getting rid of unwanted carbon dioxide.
Stop your running for the time being and check your resting heart rate several times a day for a few days. Then take the average. Yours is probably more than 80 bpm.
The next step is to get an accurate heart rate device and check your heart rate recovery. Run until you get a reading of about 85% of HRM and stop running immediately. Remain stationery and monitor the drop rate. The first minute drop should be more than 12 beats and after 2 minutes the total drop should be more than 22. Any reading less, get medical advice before any more running. An example is your high reading is 170. After 1 minute rest it should be lower than 158 and after 2 minutes the reading should be lower than 148. The faster the recovery rate the fitter your heart is.
If you find that your recovery rate is within acceptable limits,then you can start your running again. This time, when you run, do it at the 60% HRM level initially for 30 minutes or more daily. If you cannot sustain 30 minutes, either start at a lower HR level or decrease the time. But you must reach the stated HR level and duration before going further. After 1 week increase by 5 beats and repeat until you reach 90% HRM. Hopefully at this final level your legs can take it.
It will take you more than 3 months to reach the final level and your latest resting HR should have dropped by at least 9 beats, most probably more. Just carry on at 80% level until you reach the resting HR level you want. I am happy at 53.