To get full access Google products you need a Google account. This does not have to be a Gmail account but can be accessed using any email address.
To get a Google account, sign up here> https://accounts.google.com/NewAccount
As of December 2011, when you sign up for a Google account from some Google products you may be required to create a ..@gmail.com userid.
If you wish to keep the Gmail address (and mailbox) but wish to associate the email address of your choice to the account, follow step 1 below.
If you do not wish to keep the Gmail address (and mailbox) but only wish to associate the email address of your choice to the account, follow step 2.
Once you have created your Google account, what you then have to do is associate the email address you wish to use as your Google account with this new account. To do this, go to https://www.google.com/settings/ and click on the edit link next to the email addresses section
On the next page, add your preferred address for your Google account. "You can use alternate email addresses to sign in to your Google Account, recover your password, and more. Alternate email addresses can only be associated with one Google Account at a time."
You will now be able to access your new Google account using this email address.
If you wish, you can also delete gmail (and the gmail mailbox) from this Google account at which point you will be asked to provide an email address of your choice which will then become the Primary email address for your account. To do this, go to https://www.google.com/settings/products and click on the edit link next to My Products
On the subsequent page, click on the link Remove Gmail permanently and specify the email address you want to use for your account