Here's exactly how hill running should work:
As you start uphill, shorten your stride. Don't try to maintain the same pace you were running on the flat. This will exhaust you and leave you depleted later. Take "baby steps" if necessary, and try to keep the same turnover rhythm as on the flat.
Your posture should be upright; head, shoulders and hips should form a straight line over the feet. Keep your feet low to the ground. If your breathing begins to quicken, this means you're either going too fast, overstriding, or bounding too far off the ground.
You should use a light, "ankle-flicking" push-off with each step, not an explosive motion. If the hill is long or the grade increases, keep shortening your stride to maintain a smooth and efficient breathing pattern.
Run "through" the top of the hill. That is, don't crest the hill and immediately slow down or pull back on your effort. Rather, accelerate gradually into the downhill. Gravity is now on your side.
As you head downhill, stay relaxed. As with uphills, don't overstride. Keeping feet lower to the ground will give you more control. Because you're going downhill, your stride will cover more ground than it does on flat land, though it should feel slightly shorter.
Touch lightly with each step and let the steepness of the hill dictate your stride rate. If you start going too fast, shorten your stride slightly until it is under control. On gentle downgrades, you might want to try leaning forward slightly to increase speed. Just be careful; leaning too much may chop your stride or make you go too fast.
Lastly, visualize gravity pulling you downhill. The momentum you gain going downhill is a wonderful source of energy as you move to level terrain or to another hill.
Here's what you need to remember: On the uphills, reduce your stride length but maintain the same stride rhythm and breathing rate. On the downhills, increase stride rhythm somewhat (in response to the downslope) but don't overstride. Keep feet low to the ground.