How to make my (55-200mm Non-VR DX Zoom) lens create sharper pictures?

I have a Nikon D3200 and I use portrait photography with my 55-200mm Non-VR DX Zoom lens. Sometime the pictures come out so sharp and crisp, others look so blurry. I'm using aperture priority and focus between the eyes. I never use a tripod cause with portraits its restricting. I'm thinking because its a non-VR that its doing this. Please help explaining this to me and some tips to make my images sharper with this lens

The only way is with a good technique in taking photos. Here's the best way to shoot to get the sharpest images:
1. Use a tripod. Many photographers use tripods even with shutter speeds that considered "safe" to use hand held. There's a difference in sharpness even with faster shutter speeds.
2. Use mirror lock up - You don't have this feature on your D3200, but it does make a difference with macro shots.
3. Use a cable release instead of pressing the shutter button while using a tripod. If you don't have a cable release or IR remote, then use the camera's self-timer mode (usually selectable to 2 secs or 10 secs).
4. A poor stance hand-holding the camera while reduce the sharpness of your pictures. Make sure your feet are spread apart slightly wider than your shoulders. Keep your elbows close to your body. Don't hold the camera out away from your face - this is a DSLR not an iPhone! Brace the camera by placing the bottom of the camera in the palm of your left hand. Don't ever grip the side of the camera. When using long and/or heavy lenses, put the bottom of the lens barrel in your palm. With your right hand grip the camera, again keeping your elbows close to your body.
5. For long exposures hand held, take a deep breath a slowly exhale while pressing the shutter button.
6. Don't poke the shutter button. You should gently squeeze it.
7. Keep your shutter speeds up to stop motion. When you have hand hold a camera, make sure that your shutter speeds are the reciprocal of your focal length. (e.g. 500mm lens = 1/500th shutter speed).

Short of replacing your kit lens, these are the best ways to improve the sharpness of your current gear.

If they look blurry, it's probably because there's not enough light, causing slow shutter speed, and possibly because you are not using a tripod.

With slow shutter speeds you might be getting camera shake, or motion blurring - or both.

To fix it, you need more light, and you need to set a faster shutter speed, open your aperture or increase your ISO, or all of these. Or use a tripod and tell your subject not to move.

There's nothing wrong with your lens. The blurring is operator error. There was a time, not so long ago, that no lenses had vibration reduction.

The other answers are fine and correct, but a fast fix is to put the camera in Shutter Priority and set a shutter speed of 1/250. You could use a slower speed of say 1/125 with a lens such as the 18-55. Let the camera set the aperture and ISO as needed. Blur is due to a shutter speed too slow for conditions, so you simply set a speed that will not be too slow. Pretty simple and direct way to solve the problem. I honestly feel most people would have far better photos if they used Shutter Priority in most cases.

Go nearer to your subject, shoot under ideal lighting conditions and make sure shutter speed is always faster than 1/60s. You may have to raise ISO if lighting is poor. Raising beyond ISO 1600 will make the pictures very noisy hence the ideal lighting condition requirement.

Tripods and VR reduce camera shake but don't stop the subject moving.

Use a shutter speed that is no slower than the one divided by the focal length.
At 55mm use a shutter speed 1/50 or faster
At 200mm use a shutter speed 1/200 sec or faster.
But for portraits you should be using 1/200th of faster any way.

Hold the camera properly - don't wave it about at arms length using live view - use the viewfinder.