Can you turn a single player world to a realm server?

Yes! As of 1.7.10, you can upload your single player world to Minecraft Realms. The non-obvious bit is that you need to create a world in Minecraft Realms first, then overwrite it with your single-player world.

First create a worldin Minecraft Realms:

1. Load Minecraft
2. Click "Minecraft Realms"
3. Click "Create World"

Once you have a world in Minecraft Realms, here is how to overwrite it with your single-player world:

1. Load Minecraft
2. Click "Minecraft Realms"
3. Click your Realms world
4. Click "Configure"
5. Click "Reset World"
6. Click "Upload world"
7. Click the world you world you want to upload
8. Click "Upload"

Yes, with the release of 1.7.10 it is possible to upload your own world to your realm.