Have you ever noticed that when you look at a portrait (whether it be a photo or a painting) the first thing you look at is usually the eyes? This is the most important part of the picture, so when you photograph someone, try to make sure you get the eyes nicely in focus. This can be quite tricky, especially if you're using a wide aperture (as the depth of field will be small) or the subject is moving, so hold down the shutter button and take lots of pictures. Some won't be in focus but some will. In digital photography, you can just delete the ones that didn't work :-)
I've posted in this blog before about how removing distractions can help make a photo better. One way of doing this, especially for portraits, is to get close to your subject. Distracting backgrounds or foregrounds are no longer a problem if they're not even in the photo, and the results can be quite dramatic. So next time you take someone's photo, try taking it as you normally would, then step closer and take another, then step closer again...
If you decide to give it a go, post your results on my Facebook page, I'd love to see them!
I think the best way to improve your photography is to practise, practise practise. One way to encourage yourself to do this is to start a project that forces you to get the camera out and use it regularly. So why not challenge yourself to take a photo every day for a year? This is something I tried in 2010/11 and I found it a lot of fun. Try constraining yourself to all landscape, portrait or square framing : it makes the poster easier to make at the end :-)