running will actually help your asthma long term... just don't go far without an inhaler and mobile!
if you can already run 1km then you can train from there, no need for walk breaks just do a couple more 1km runs then increase to a couple of kms and so on... if you can't complete it then slow down a little bit, 5 mins for a km is a pretty decent pace if you maintained it over several miles so slowing down a little for runs where you're increasing your distance is fine