Let me try to sum up what you have to do:
First, you need to buy a PSTV, which is rather cheap.
A PSTV is basically a Vita with an HDMI out instead out a screen and Bluetooth to connect with PS3/4 controllers rather than having buttons on it. It can't run all Vita apps/games, for further information about which games it can/can't play see these pages:
https://en.wikipedia.org/wiki/List_of_PlayStation_Vita_games
http://vitareviews.net/playstation-tv-compatible-games-north-america/
If you have experience with capturing videos from HDMI then you should know the rest.
If you don't, you're just gonna need an HDMI recorder/capture device, ten seconds of google gave me those two
http://www.amazon.com/Elgato-Capture-PlayStation-gameplay-1080p/dp/B00840353W
http://www.amazon.com/JIAFENG-100-Digital-blue-Ray-Recorder-PlayStation/dp/B00KVCGV5K
I'd advice further research before getting one.
Edit: if you aren't in a hurry, I'd also advice waiting for one that supports HEVC/H.265 encoding, which if you don't know basically doubles the video files compression rate (halves the size) while having the same video quality. Some people might say that won't be happening anytime soon, but with Microsoft, Apple, Google, AMD, nVidia, and Intel supporting HEVC/H.265 in their latest releases, I doubt it'll take too long before it becomes the norm.