I am trying to add an additional outlet in the middle of a series of outlets, this is in the kitchen to run indirect lighting in cabinets. Since the existing outlet is in series how do I connect into it? Do I splice into wires or do I disconnect one set of wires from the existing outlet and join the new outlet with the old wires and piggytail it in?

Your outlets are not really in series, electrically anyway, physically yes. As long as the additional load will not increase much, you should be ok by adding the lights. You can "pigtail" off of an outlet up to your lights as long as you keep within certain criteria. i.e. don't overfill the boxes, make proper connections, etc. There are rules to follow on how many wires (and other things) in boxes, however, if you have to stuff things in, then there is a very good chance you have overfilled the box, not a good thing. Just attach hot wires (black usually) to hot wires, neutrals (white usually) to neutrals, grounds to grounds and you should be ok. Of course, this is simply a general observation and if you are not sure of procedures, it might be a wise investment to have a qualifed electrician do the work. Good luck!