Recording Full HD 60fps

Do you have a separate HDD dedicated for recordings? You'll need one if you want to have smooth recordings, trying to record to the same HDD you're playing a game off of is what causes those hiccups usually. (Games being picky for stupid reasons can cause them as well)

You can try OBS since it's free:
https://obsproject.com/
And here's how to set it up for local recordings, start at the recommended crf 15, and lower the number for higher quality and higher file size recordings, or raise it for slightly lower quality and smaller file sizes.
https://obsproject.com/forum/resources/how-to-make-high-quality-local-recordings.16/
And to set it up for Twitch:
http://help.twitch.tv/customer/portal/articles/1262922-open-broadcaster-software
And to get it setup for Youtube streaming:
https://support.google.com/youtube/answer/2853700?hl=en&authuser=1&ref_topic=6136989
And optimizing it for Youtube:
https://support.google.com/youtube/answer/2853702?authuser=1

If you have an Nvidia GPU, you can use that to reduce the load on your CPU:
To set it up for recordings:
1. go to settings
2. go to encoding
3. click the Nvidia NVENC button
4. Make sure Use CBR is turned on
5. Make sure Enable CBR Padding is turned off
6. Set your Max Bitrate (Kb/s) to your upload max upload speed. You'll want like 8000 or more though, this affects the visual quality of the video partly. (1mbps = 1000 kbps, www.speedtest.net)
7. go to Video and set the FPS to 60
8. Go to Advanced
9. Turn Use Multithreaded Optimizations on
10. Set NVENC Preset to High Quality
11. Set Encoding Profile to Main
12. Turn Use CFR on.

To capture your game:
1. Right click in the white area of the Scenes: box and choose add scene, name it after the game you're gonna play.
2. Start the game you're gonna play.
3. right click in the Sources: box and choose Game Capture (for full screen games) or Window Capture (for windowed games whether borderless or not) or Monitor (to capture whatever is display one your monitor including your desktop/webbrowser/OBS)
4. Start recording.

To add a webcam:
1. Right click in the Sources: box and choose Video Capture Device.
2. Choose your webcam from the dropdown list at the top. Press Okay.
3. Click Preview Stream
4. Click the Video Capture in the sources list, go to Order and choose Move to Top.
5. Press the Edit Scene button

No, the exterior ones will be too slow, and will also result in choppy recordings, but are fine to store the files after you've recorded them.

Because video files are so big, an SSD isn't very economical, unless you want to transfer videos all the time.
You only need a 1tb 7200rpm internal HDD for recording videos, and that only costs about $40-45, an SSD for the same price is going to have a quarter or less of that space.

You may also investigate using a program like ProcessLasso to assign separate CPU cores to CSGO and your recording program so they don't try to compete for the same CPU cores and thus reducing any hicccups as well.