How to wright a proposal of a web design to a company

Generally you would use use cases to describe standard interaction patterns between a typical user and the website. For example, use comes to the site, searches using a keyword which she types into the search box. The site then responds with a paginated result of links that give the user a quick overview of the kind of site it is. If the user clicks on a site link they are taken to that link in a new tab...

You can see that the above interaction pattern is the typical use case for a Google user. Likewise, do the same for your idea/site and write it out.

Additionally, you have to get visual. If you can express yourself simply with boxes and arrows you are drawing the information architecture using a wireframe. This is also a very good means of communication of the desired information, interactions items and "call to actions" on the site. If you are doing a graphic design proposal, I would use Photoshop to mock something up that encapsulates the above.

In my experience, these are generally the tools I have used received from design firms when they are proposing a site design. Of course there are plenty of other approaches but these are pretty standard.

This is a big topic but there are tons of resources out there. Good luck with your proposal.