The best way to build up your running fitness is, surprise surprise, to start running! And the best way to do this when first starting out:
Combine short bursts of running with easy bouts of walking.
No matter what your current fitness level this beautifully simple combination, popularised by running coach Jeff Galloway, will allow you to get started. As your fitness improves you will quickly discover you can run for longer and longer before taking a walking timeout.
Guidelines:
Run at an easy pace until you begin to tire
Walk until you are ready to run again
Repeat this combo for a time or distance of your choosing
If you are just getting going start with really small targets, like 5 or 10 minutes, and steadily build up from there. Among other things this will allow your connective tissues (which adapt at a slower rate than your muscles) to catch up with the extra demands running will place on them. Too much, too soon is the downfall of many a rookie runner!
Benefits:
By integrating periods of easy walking into your runs you will be able to effectively manage your fatigue, get more enjoyment from your early workouts, avoid burnout, reduce the risk of injury and maintain a higher level of motivation... all while improving your fitness.
It really is perfect for those just starting out. Give it a try and see!