You want to have your kids portrait, but sometimes its really hard to get it, especially when kids so active and always move around. I have a few tips for you.
1. Catch the moment!
Try catch the moment when they in right place and right time! Its not so easy, but you can do it (just watch them playing and moving around and have your camera ready with fast shutter speed :) And when you see this special moment just made a click :) Better few in continuous mode for be sure you get it right :)
2. Create this special moment
Take kids attention -
smile to them and they smile in answer,
talk to them and take their attention, ask some question - where the bird? and sure they interested look at window or tree and show you the bird,
made silly faces, jokes - made them have fun and laugh.
ask them to make serious face - and you get the most funniest faces ever!
made something interesting for them - flash works good with kids, its a magic for them, so make this magic around and they will be happy repeat this moment, also you can took their attention with mobile phone or computer (and get more light from screen :)!
play with them - made for kids interesting task - jump as high as they could, run as fast as they can, made a secret, choose one person in family and jump in the hug, draw something, made mess (they love it!) etc. I use a lot of games and play moments during photoshoot and its always work well with kids, they just having fun and show their personality :)
3. Look for light - be sure your kids face in good light (without hot spots) and eyes had some catchlights, its looks really good. Half face could be in shade (if light goes from the side like window light) and its made more dramatic look (especially with black and white conversation).
4. Keep kids eyes in focus - be sure your focus right, check on eyes if they clear to see without blur.
5. Play with aperture - you could make blurry background with small aperture, just be sure your kid far from background (otherwise its will be same clear like kids face).
6. Watch the composition! - try get it right, so picture take attention (you could use rule of thirds, repetition, lines moving to subject, colour attention, isolation, using negative space etc. - its a lot of things to count but the main thing - picture just need looks right for you, all in harmony!
7. Just enjoy the process and have fun! Only if you relax kids relaxing and having fun too, be sure you made for them interesting environment and they even not put attention on camera.
It will be great results!