The absolute first thing that runs as the Web is not your application, but the method OnStart()in WebRole.cs in your Azure project. This is the place to add code to monitor your website.
The standard method is to copy application trace logs and Windows event logs to the Azure table storage together (if necessary) using tools to use CPU, IIS statistics, and what you have.
A good introduction to this is here: http://blog.bareweb.eu/2011/01/beginning-azure-diagnostics/
as well as a detailed description of the features that you will need in your application: http://blog.bareweb.eu/2011/03/implementing-azure-diagnostics-with-sdk-v1-4/
which remains applicable for Azure SDK 1.5.
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