Can i request a new log book from the DVLA?
you need a V62 form and a fee of £22 sent to the DVLA to apply for a new registration document in your name. Catergory C is not the end of the world, just means the car was deemed as uneconomical to repair by an insurance company. It is not illegal to sell this car, and because you bought it privately you have no legal come back on the seller. You would have to prove they knew it was a write off and failed to inform you of this. Just get the car checked to make sure it has been repaired correctly, also bear in mind it could be accident damaged, stolen recovered, flood damaged.