I'd also love to see this capability, but it seems from this MSDN page that it's not possible. I hope they expand the service's capabilities later on as it would be great to see a product on the level of AppDynamics or New Relic from MS.
Performance and exception monitoring with Application Insights
You can monitor performance in:
A .NET web service. You have to have administrative access to the server machine. You don’t have to alter the code of the web service.
A .NET service running as a web role on Microsoft Azure Cloud Services. You have to have access to the code. You run the Application Insights tool on your development machine, then rebuild and redeploy the service.
A Java web service. You download the agent to your server machine, update the configuration files, and restart your web service.