Looks like no one wants to help you so I will. First of all, do you want to play on your TV, because you can just plug the ps3 directly into the EasyCAP and record like that, the audio is a little fked up though.
Beware buying an EasyCAP though, there are alot of fakes. I think mine might even be one. [ame=
https://www.amazon.com/EasyCAP-DC60-Creator-Capture-High-quality/dp/B002H3BSCM/ref=sr_1_1?ie=UTF8&qid=1293409547&sr=8-1]Amazon.com: EasyCAP DC60 - USB 2.0 Audio/video Creator Capture High-quality Analog Video: Electronics[/ame]
If you're going to get an easycap, you'll have alot of problems and unless you're tech saavy, youll have trouble setting it up on your PC. The drivers just DONT work at all, so youll need to find some modified ones. I'd recommend a daazle, but you'd be about 10 - 15 dollars over you're $50 budget, if you are trying to play on your TV while recording.
[ame=
https://www.amazon.com/Pinnacle-Systems-82303000101-Dazzle-Recorder/dp/B0039Y2YDK/ref=sr_1_cc_1?ie=UTF8&qid=1293409693&sr=1-1-catcorr]Amazon.com: Dazzle DVD Recorder Plus: Software[/ame]
If you don't have a Laptop youll need an AV extension cord.
[ame=
https://www.amazon.com/RiteAV-Audio-Video-Cable-50ft/dp/B000V0DY8Y/ref=sr_1_6?s=electronics&ie=UTF8&qid=1293409799&sr=1-6]Amazon.com: RiteAV Audio Video RCA Cable 50ft: Electronics[/ame]
You might not need one this long but you get the idea.
Next you'll need 3 splitters
[ame=
https://www.amazon.com/Cables-Go-40650-Female-Adapter/dp/B000M52X62/ref=sr_1_1?s=electronics&ie=UTF8&qid=1293409832&sr=1-1]Amazon.com: Cables To Go 40650 RCA Male to Dual RCA Female Adapter (Black): Electronics[/ame]
You'll need some female to female AV depending on what you buy, because males don't connect to males, obviously.
[ame=
https://www.amazon.com/Female-to-3-RCA-Coupler/dp/B002B8WVVU/ref=sr_1_1?s=electronics&ie=UTF8&qid=1293409904&sr=1-1]Amazon.com: Female to Female 3-RCA Coupler: Electronics[/ame]
So you take your ps3 cable, attach a splitter to each end.
---[ps3 cord]----[splitter]<
attach your AV extensions [Youll need 2] [ame=
https://www.amazon.com/RiteAV-Audio-Video-RCA-Cable/dp/B000V0DY70/ref=sr_1_6?s=electronics&ie=UTF8&qid=1293409988&sr=1-6]Amazon.com: RiteAV - Audio Video RCA Cable 3ft: Electronics[/ame]
........................................[extension]
----[ps3 cord]---[splitters]<-
........................................[extension]
Now plug in 1 extension into the TV, and the other into the Dazzle
........................................[extension]-----[Dazzle][][PC]
----[ps3 cord]---[splitters]<-
........................................[extension]-----[TV]
Very crude but you can kind of get the idea.
It's alot cheaper to play through your PC, but it can lag, and throw off your game. Playing through the TV is a little bit more complicated, but you play with no lag, and on a bigger screen possibly. Watch a set up video, it explains it a little bit easier to understand.
Good luck!