Well there are a few options with your existing lens.
What you haven't told us is what shutter speed and focal length you're shooting your portraits with, when they have this problem.
As a rough guide, take the reciprocal of the focal length as an estimate of your minimum shutter speed to get a few more keepers and less subject blur.
For example, if you're shooting at 60mm, then 1/60th second is the minimum you can go to in order to get reliable keepers hand held. You can go slower with vibration reduction, but then your subject may move slightly anyway, increasing the chances of blurring.
If the light is not good enough and you're shooting without a flash, handheld, then you can up the ISO in order to hit your minimum shutter speed.
For example, if you're shooting at 1/30th second, ISO 200, at 60mm and want to hit 1/60th second shutter speed, doubling the ISO to 400 will get you to 1/60th second.
Even with a kit lens, you will be able to get seperation between the subject and the background. It's all to do with how close the subject is to you, relative to the background and focal length. Even at F5.6, you can get a blurred background.
However, to get really blurred backgrounds, it can help using a faster zoom (expensive) at F2.8 or a cheaper prine lens (e.g. 35mm F1.
at wider apertures.
However, even with the most expensive lenses (with a few very expensive exceptions), you will get maximum sharpness and contrast out of any lens when stopping down around 1-2 stops from the lens' maximum aperture.