The best AMD chip for video transcoding is going to be an AM3+ chip
Link But there are about a dozen Intel chips that are faster, most cost more money though.
A good pairing of the motherboard would be
Link This does have an HD3000 integrated graphics, so you wouldn't need a video card, but you may want one as the HD3000 is pretty weak. Not a lot to choose from in mATX boards, only about 10 at Newegg. If you were to step up to a full ATX, and it would fit in the case you might want to look at
Link, But this does not have a video card. You could use a video card you have, or an R7 260x card is decent for the price
Link.
You also need to consider a quality power supply, not knowing what is in your old system,
Link This Corsair is a great buy. I have 4 Corsair power supplies in my systems at home, and they are great.