Callsign includes the term 'with Kilo' ; What does that signify?
What that probably means is that the pilot is reporting that they have ATIS information kilo. Automatic Terminal Information Service (ATIS) is a radio broadcast on a specific frequency (often a local navaid like a VOR) which a pilot dials up to get weather and airport information before joining one of the airfield's frequencies. Each time the information is updated in a day the next letter of the alphabet is used; "information Kilo" means the information has been updated 11 times that day. 

If the tower had to relay weather, runway in use and other instructions to everyone pilot who joined the frequency it would take loads of their time; ATIS is instead there so controllers can spend their time controlling. When a pilot introduces his/herself on frequency at an airfield they announce the latest ATIS they have so that the controller knows they have up to date information and they don't need it relayed. 

From a phraseology standpoint just saying "with kilo" is okay. I usually say "with information kilo" because it makes sure that kilo isn't confused as being part of my callsign.