Proper way of running

[Conscious]

Slightly off-topic, you run differently barefoot then you do with a shoe on.

Back on-topic, it depends OP. If you're sprinting you want to lean forward and move your feet literally as fast as you can. If you're going for distance it's about strides, striking, rolling through, and some more strides.

[Kreyzor]

For sprinting, at least relatively short distances, striking with the ball of your foot gives your the most force and propulsion and is generally the most natural sprinting stride.

As for distance or jogging, I see a lot of people saying heel striking and rolling your foot from back to front is the way to go, but from what I've experienced and seen that's generally not true. Mid-foot striking is considered the most efficient way to run, and can take some practice to get used to. If you watch in slow-motion pretty much any professional runner you'll see they have a mid-foot strike with their foot landing directly under their center of gravity.