First of all, you need to get a Gateway. You can do so at the Black Market by tapping on the Misc. tab. You can then apply it right away by tapping on the Apply button, or you can save it, finding it in the My Items tab of the Black Market later. Then, once you're ready to relocate your Castle, tap on the Apply button and follow the instructions.
You will also receive a report on your relocation attempt from your Scribe! All you have to do is select Relocation in the drop list to check the report. Note that in case of relocation failure your Gateway will remain unused, allowing you to use it at a later time.