You must win 100 single mode (1 v 1) matches. I can confirm that both single mode (playing against the AI), and multiplayer single mode matches count towards the achievement. I unlocked my Lonely Duelist achievement from a single mode match playing against the AI. One of my friends unlocked his achievement last weekend in a boosting match.
Now, the easiest way to do this achievement is to boost it. Have one player create a Player Match with a 30 second timer and a private slot. That person should invite the other person to their game. The game will start and the player who wants to win the match will choose Rock in Rock,Paper,Scissors. The player that will lose should choose Paper. Then, the player going first should let the time run out. I recommend doing this in intervals of 10 matches won. Then, switch to the other player.
Now, you can also get this by playing legitimately. But, this will take much longer. I suggest that you use a deck that is based around either "Blackwing" or "Lightsworn" cards. These cards seem to be the best. And if you can't find an online match, you can play single mode against the AI. I suggest that you only play the first 3 matches. These are the fastest as they are each only 1 duel long. Also, you can quit out of the match if you think that you're going to lose. The matches you win will count. For starters, I suggest the H.S. character deck because it deals much damage and you can usually get around 300-400 points each match with it.