The obvious problem with your sample photo is that it's blurry. There are several factors that can contribute to the problem, including:
- camera movement
- subject movement
- out of focus
- poor optics
- sensor noise
- shallow depth of field
When you're taking a photo with low light, you'll generally be working with a large aperture, which means shallow depth of field, and a relatively long exposure, which increases the likelihood that either the camera or the subject will move. If you try to combat these by increasing the ISO (sensitivity), you'll increase the noise in the photo. Also, the camera's autofocus system may not work as well if you don't give it enough light.
Things you can do to improve your images with your current camera:
Stabilize the camera. A tripod is one way to do this, but it's not the only option. Simply placing the camera on a tabletop rather than hand holding it would be a good start. You could use some zip ties to attach it to a brick or other heavy object.
Smaller aperture. If your camera lets you set the aperture manually, set it a higher f-number (higher f-number means the light passes through a smaller hole). This will increase the depth of field, meaning that the range in which the subject is in focus increases. The camera will have to use a longer exposure to compensate, which could lead to motion blur, but if you've stabilized the camera as suggested above this doesn't have to be a problem.
Lower ISO. If you can set the ISO manually, choose a value between 100 and 1000. The small sensors in most point and shoot cameras generally don't handle low light very well. Again, this will lead to a longer exposure, but that's okay if nothing moves.
A different camera will help a lot. You don't necessarily need a DSLR (digital single lens reflex) camera, although that will go a long way toward solving your problem. Look for something that at least lets you focus manually and has a significantly larger lens. The lens on a DSLR collects a lot more light than a typical point and shoot, and also has a larger sensor for recording that light. You'll get manual focus, generally better low light sensitivity, and better optics. You don't necessarily need the latest and greatest to realize these benefits -- a used camera in decent shape will give you all these benefits at a price that's more likely to fall in your budget.