Little late now but never a great Idea to start with a drain cleaner.
Just remove the trap under the sink and clean that out, if it's clean then leave it off and remove the pipe sticking out of the wall and run a snake down the elbow behind the wall.
Be very careful since you already have dumped a caustic liquid down the drain.
Need more info, what size and type of snake? How badly is it clogged? Do you have a garbage disposal?
Off hand, I would remove the trap, snake the line as far as your snake will go, and then follow that up by flushing the pipe with a bladder such as a drain king. That usually does the trick for most of our units, but we're using a powered snake.
It's a kitchen sink drain, so odds are it's grease that's clogging it, if you still can't get it you might call around to see if a proper drain cleaner can jet the line.