You'll need to download each image referenced (and CSS and JS and any other referenced object you want to count). Some of them may tell you the size of the object in bytes at the start of the response, allowing you to get just that and then close the connection.
To find the size of the root HTML itself, simply count the characters as you read them from the stream.
If this doesn't answer your question, please post a more detailed description and -- even better -- code.
I’ve looked at a lot of different solutions, and in my humble opinion Aspose is the way to go. Here’s the link: http://aspose.com