€550 Streaming PC build

ConorK95

Hi guys, I'm a first time participant on the forums but I've been reading for a few years, so I know I'm putting this question into good hands.
I'm looking to build a decent pc for streaming on twitch through OBS. I know it's not much, but the €550 budget is all I have to work with, maybe pushing it to €600 at a real stretch, as I am just a lowly poor engineering student.

1. What is your budget? €550

2. What will be the main purpose of the computer?
Live streaming on Twitch. Mainly PC games but may move on to capture card PS4 after a while (although there's no need to worry about that now.)

3. Do you need a copy of Windows?
No.

4. Can you use any parts from an old computer?
No.

5. Do you need a monitor?
No.

5a. If yes, what size do you need.

5b. If no, what resolution is your current monitor and do you plan to upgrade in the near future?
1920x1080
Not anytime soon.

6. Do you need any of these peripherals?
Wireless Card. Also, I'd need 2xVGA ports or one VGA and one HDMI

7. Are you willing to try overclocking?
I'd prefer not to.

8. How can you pay?
Preferably Visa or Laser, or Paypal. I don't have the time to make it to a bank to set up a transfer.

9. When are you purchasing?
Within 2 weeks hopefully

10. If you need help building it, where are you based?
Galway City

Thanks for all your help!

ConorK95

Some extra info:

I tried to do a test stream on my laptop, but it's not powerful enough. There were problems with high encoding on OBS, and some research has lead me to believe that is mainly because of the integrated graphics card not being powerful enough