I have a Fuji S5100 camera with the macro function, but have a terrible time taking closeups. Here's an example:

I took this picture from about two feet away. I tried getting closer, farther, zooming in and out and this was the best I could do. Is it my technique or maybe just a camera with a lousy macro function?

Try a tripod, and the self timer.
You should be able to get closer than 2 feet with the macro.

I used both a tripod and the self timer. And also the camera was set to auto mode.

Are you partially depressing the shutter button to set the auto-focus before depressing it to start the self timer?

Even with my one camera without stabilization, I can get closer than two feet.

No, I didn't do that. I'll try another shot and see if that does it. Tthanks.

In addition to what Dave said, use a darker background. The camera wants to average everything and that will make the dark interior even darker. Since you're already using the self timer and tripod, add at least one other light source to illulminate the interior. You could use a flash light or a desk lamp with adjustable neck.
Make sure your spot meter, if you have one, is engaged. There will be a cross hair in the middle of the viewer to let you know where the cameral is "looking."

Here's another pic after taking Grumpy and Al's advice. A little better, I think. Thanks guys.

Sorry for the late reply Cary.
Yep, the last photos certainly is better.