I have problems playing FullHD 1080p MPEG-4 AVC videos on my desktop and on my laptop, so it should be pretty hard to record FullHD videos using software.

Is there any reason to record FullHD 1080p videos? I suggest you to reduce screen resolution and record using Huffyuv or Motion JPEG video codec. If you don't use any compression when recording from you screen, your recordings will be huge. You can then recompress your recordings using x264 (MPEG-4 AVC) or xvid (MPEG-4 ASP) video codec. I suggest you to use CRF encoding to get better quality. 2-pass ABR encoding gives worse quality than 1-pass CRF encoding.