What sort of specs will I need for a desktop PC which can do all this?

At the moment, I'm looking to buy a desktop computer due to the fact that the one I have now doesn't cut it anymore. The main things that it will need the power to do will be: Record video using fraps, play Minecraft on high/full settings, and use skype. These three things will sometimes need to be used all at the same time, which my current PC can't do without lag.

I have thought about maybe building my own computer, but I feel like it might be too difficult to build, and I don't wanna risk breaking any of the parts. If however, building it will be a lot cheaper, then I may consider it.

It isn't hard to build a computer and following some simple rules about handling hardware and you would be fine. Building your own has the advantage of allowing you to get exactly the parts you want. The disadvantage is that you don't get a factory recovery image and you have to purchase the Windows license separately.

Look at this system as an example:

https://www.alternate.nl/ALTERNATE/Upgrade-Kit-ASRock-970-Extreme3-FX-8350-8-GB/html/product/1075231?tk=7&lk=9527

Building your own will be a very rewarding learning experience, I used to build lots of them and then sell them.
My most recent buy was a 17" hp notebook with an i7 cpu and 16gb ram. I'm finding windows 8.1 is NOT the ogre we have been led to believe.

Building your own computer is a lot cheaper, and a lot easier to fix and upgrade when it needs it. I wouldn't classify it being hard, but there are definitely a few things you should remember. Things such as, don't touch the tips on the CPU, or on the Motherboard (can't remember which brand, but I believe its AMD's mobos and CPUs where the tips are on the motherboard, might be Intel) don't touch the screw driver right onto the hardware such as the RAM.
Look through this site
http://www.computershopper.com/feature/75-pc-building-tips
http://lifehacker.com/...1458484291 that site is a bit shorter, but still helpful. One thing I would say though about the CD's or DVD's don't throw 'em away.
To run Skype, a desktop will need a Webcam. Logitech makes really good ones, and most shouldn't break your budget, or if you have one, simply use that.
I don't know how much you are able to spend on a computer, so that information would be a tad helpful.
A CPU cooler shouldn't be needed, unless you are over clocking, as the ones that come are usually good enough, even for gaming.
http://ca.pcpartpicker.com/p/trr623 This should suffice.

Don't down yourself. Building PCs (IMO) is like making a sandwich, you have to have certain ingredients to make it the best sandwich youve ever had! Please update your question to where you specify a budget, otherwise nobody can truly help you.

here are the specs of my first PC that got me averaged 100-175 FPS on MC MAX settings:
CPU-AMD FX-8320E
MOBO-Asus Sabertooth 990FX R2.0
Memory-G.Skill F3-17000CL9Q-16GBZH (16Gb 4Gbx4)
Storage-Seagate ST31500341AS
GPU-PowerColor AX7850 2GBD5-2DHE/OC
Optical Drive-Samsung SH-224DB/RSBS
OS-Windows 8.1 Professional 64Bit
PSU-SeaSonic SSR-550RM