Do I need anything from Apple on my computer if I am running Windows on a HP?
You don't really need anything from Apple, but they have got some good software you can use on Windows. You could download their browser Safari. It's not as good as Chrome in my opinion, and it is not updated anymore, but you might like it.
If you have an AirPort Express/Extreme- or a Time Capsule-based network, you will need the application AirPort Utility to configure it.
I do recommend you install iTunes, Apple's music application. It's pretty good for playing music, and if you have an iPhone or iPad, you can synchronize and back it up with your computer using iTunes.
If you're using iCloud, you might want to download the iCloud Control Panel, allowing you to sync your bookmarks (with Safari) and your Photo Stream, in addition to contacts, calendars, tasks and mail with Outlook (requiring that you have Outlook on your pc).
Those are all the applications I can think of that you can have any use of (except maybe QuickTime).
Harware-wise, you have a lot of options. You could get a WiFi-router, such as AirPort Extreme/Express or Time Capsule, or EarPods (for music), and so on. Take a look at the online Apple Store for a full overview.