what is the brand of this sink faucet?
just looking at the surface and general look and fit and finish, it looks like a made in china junk faucet.  these types of faucets are essentially disposable.  even if you can get it apart (which may not be the case), you will never get parts for it.  

if you do want to try it, try removing the supply line and nut from the bottom (under the counter) and lift the handle assembly out.  there may be a set screw or something under the escutcheon.  i have seen enough of these to not offer much hope though.  get yourself a moen faucet as a replacement.  good pricing, lots of choice, and free parts for life.