That is something your IT department will have to handle. They will need to add a "friendly" DNS name for your PC to their internal DNS, rather than the "unfriendly" name it probably has now. (Like "Engineering-XU4583").
Other than that, there's nothing you need to do to "deploy" your server within your company, other than to make sure your machine is on and the web server is running.
(OK, perhaps add a link on the company's "main" Intranet server. But, again, that is something for your IT department, or whoever maintains that server.)
I don't see anything wrong with using your PC - at least initially - to provide a web service within your company - as long as the users understand that it's an experimental service that might be subject to interruption.
Do be careful of politics, though. Some might feel threatened by your action. I usually recommend "it's easier to ask for forgiveness than for permission." But as a student intern - maybe better to ask permission first. :)
Anyway, you may find people find all sorts of excuses why they can't/shouldn't help you. Hopefully, though, they will be delighted and cooperative. Welcome to the business world!