I don't recommend upgrading until you can afford a better CPU.
Some raw data:
http://www.cpubenchmark.net/cpu.php?cpu=Intel+Core+i3-6100+%40+3.70GHz
https://www.cpubenchmark.net/cpu.php?cpu=AMD+FX-8350+Eight-Core
(single thread performance benefits the i3-6100 thus most games. multi-thread benefits many/most video converters)
1) i5-6400, i5-7xxx (Kaby Lake) or better
- problem here is that you now replace the motherboard and system memory
2) i5-4590 or similar
- need new motherboard but not system memory (depends on how much DDR3 memory you already have)
3) ZEN?
- if you can wait, I'd see what ZEN has to offer in 2017
- will need new motherboard and DDR4 memory though
Summary:
So the main problem is adding the motherboard, possibly memory. I think Windows 10 is transferable but W8 and prior are not so that's another cost, as well as REINSTALLING all your software.
Unfortunately, for most games AMD CPU's are a bottleneck which can drop performance as much as 40% or so vs a good Intel CPU but that varies. A lot.
So again, considering your budget appears tight now it's hard to recommend the Intel path especially when the FX-8350 is faster by up to 65% vs the i3-6100 (Handbrake is a great program and can use about 100% of the CPU's threads thus just divide 8938/5418 for a close estimate).
I would strongly suggest holding out for Zen, and possibly investigate used parts.
Other:
I don't know how good "Convertxtodvd" is however I find using DVDdecrypter (or similar) then using HANDBRAKE works very, very well. Handbrake is very good for tweaking the quality vs size/time for the video. I'll list some of the important settings in a post below.