There are good and bad points, but the most telling way to judge him is based on previous season. Here is how stood after 23 games for the last 4 seasons:
2012/13 BR:
Goal Difference +12
Points 34
Position 7th
2011/12 KD:
Goal Difference +7
Points 38
Position 6th
2010/11 RH (plus a game or 2 of kenny's):
Goal Difference -1
Points 29
Position 10th
2009/10 RB:
Goal Difference +14
Points 38
Position 5th
So, over the last 4 years, Rafa (the season he got sacked) had done the best after 23 games. Rodgers can only be described as doing 'not very well'. That said, the league has become more competitive in the last few years so I'd upgrade BR slightly and say he's doing just about enough to keep his head above water. Only Hodgson did worse. Big improvements needed from Brendan though.