Whoa. It sounds like you're either an intern, or a very young developer hired just out of college, with no idea of how the corporate world actually works. You're soon in for a very nasty surprise if you continue down this course of action.
As multiple other people have probably told you, what you're trying to do is almost certainly against your company policy, and almost certainly will get you fired if you persist. I mean, you're playing with your career here. Once you get fired with cause, it's nearly impossible to get hired again in the IT world.
You need to stop trying to route around the procedures and policies in place at your workplace, and start working with your coworkers and management to convince them that what you're trying to do is a good thing for the business. If they ultimately tell you "no" and you still believe you're in the right, you can then start polishing your resume and try to find another place to work that's a little more open-minded.
The policies at your workplace are there out of an overabundance of caution. I'm not saying the policies are the best thing in the world, but they are approved by the management, and if you violate those policies, there are very real consequences for doing so that might require you to change your career.
Yes, it will take longer to buy the hardware/software you need through the company's official channels. Yes, the system administrator might give you a hard time. But ultimately, if everyone else in the business knows what you're doing and fully supports it, everyone will be better off. They might even change the policy to allow similar things to happen more easily in the future, if your effort pays off.
Blindly telling us in your comments to "just give you the answer" is the most inane thing you can possibly do. One, if you get it to work, and you're caught or detected to be basically a rogue operator, you're probably going to get fired.
And two, it's not technically possible based on the way you've set it up now. You could install a second network card in your main (official) company desktop computer and use Windows' built-in Internet Connection Sharing to give your other box local network access, but this would be a further violation of your company's policy on not modifying their equipment or reconfiguring their software.
This whole cowboy "I'm better than the policy" attitude is really playing with fire. Step over the wrong boundary and you'd better learn how to flip burgers. This is the most correct answer to your overall problem that I can give you: work within the system to change and mold it to your needs; don't try to bypass the system. If you do, you'll quickly be branded a rogue operator and you will never be able to get another decent job in your life. Despite not knowing you at all, I wouldn't wish this fate on you. Wisen up.