To try Azure preview features go to the Azure Preview site and click Try it next to the feature that you want to preview.
You can see which Preview Features your account is currently signed up for by signing in to https://account.windowsazure.com/PreviewFeatures
In the Azure Portal you can also hit the N key on your keyboard to bring up the New blade, and type in preview, this brings up a list of services, some of which are not listed on the Azure Preview site.
If you want to preview features specific to just the portal interface, just released is the Azure Preview Portal which you can access by going to https://preview.portal.azure.com/
Check out Chris Pietschmann's blog post about it
To submit new ideas for the portal, click the drop down menu next to your account and select Submit an idea
If you would like to find out more about our Azure services, please get in touch with us.
I have written a sister blog post to this one that covers Office 365 preview features. Please check it out