How can I create my own (new) alliance?

You can only form an alliance if you are currently allianceless. To create an alliance, click on the "Alliance" tab in the Games menu and then on "Create Alliance".

If you are still in an alliance, you must first click on the "Alliance" tab in the menu and then on "Overview". At the bottom left of the table you will then see the red button "Leave Alliance". After you have left your old alliance, you can form a new one (see above).