Customizing the Office 365 Public Facing Website

For Customizing the public-facing Website that is available with SharePoint Online for Microsoft Office 365 you should go for following options:

Option 1: Out-of-Box Editing

Option 2: Apply Minimal Master Page and customize it using SharePoint Designer

For customizing website through Out-of-Box you can refer this Article : http://office.microsoft.com/en-us/sharepoint-online-small-business-help/step-3-customize-the-public-facing-website-HA102029499.aspx

When you try to customize the website master page through SharePoint Designer 2010 it will gives you different errors. So to avoid these errors firstly copy Minimal Master Page (Create duplicate file) which is located to /_catalogs/masterpage/minimal.master, then customize it in SharePoint Designer 2010. Apply this customized Master Page to all pages.