How do I close my own question when I don't get any answers?
If it's a good question you should probably just leave it, but if you really want to get rid of it you probably want to delete it, not close it. From the FAQ entry about deleting posts:
Since it has no answers, you should be able to delete it yourself; there's a link just under the tags.
There's no reason to close or delete a question just because it hasn't gotten any answers. A better approach would be to edit the question to provide more details, making it more likely that you'll get an answer. But even if that's not possible, it doesn't hurt anything to leave it open. Someone is likely to come along at any time who knows the answer. If it's closed (or deleted), they won't be able to answer it.
You definitely should not close/delete the question and ask a new one. As mentioned above, editing is the appropriate course of action in this case, and it still "bumps" your question up to the top of the list.
If you still decide that you want to close/delete the question, you have a couple of different options. As Michael suggests, questions that have no answers can be deleted by the original poster. Just click the "delete" link underneath the tags. Alternatively, you can flag a moderator (using the "flag" link), explain your reason for wanting to close/delete the question, and ask them to do it for you.