PREMIER 1A FLT CREW HAS A TCAS ALERT DURING DEP FROM MDW.

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OUR INITIAL CLRNC FROM CLRNC DELIVERY WAS AS FOLLOWS: MIDWAY 4 DEP; AS FILED; CLB AND MAINTAIN 3000 FT; DEP 133.5; SQUAWK XXXX. I READ THIS BACK AND WAS TOLD READBACK CORRECT. WE TAXIED TO RWY 22R. OUR CLRNC FOR TKOF WAS AS FOLLOWS: CLRED FOR TKOF RWY 22R; FULL LENGTH; R TURN TO 250 DEGS. THAT'S WHAT I READ BACK. WE TOOK OFF; TURNED TO A HDG OF 250 DEGS AND WERE JUST PASSING 2000 FT WHEN I NOTICED ON THE TCAS AN ACFT APPROX 2-3 MI; 12 O'CLOCK POS; AND 400-500 FT ABOVE. I NOTIFIED THE PF TO DSND. AS WE WERE DOING THIS; TWR TOLD US TO MAINTAIN 2000 FT. WE DID AND THE OTHER ACFT PASSED ABOVE AT 400-500 FT. TWR SAID HE ISSUED 2000 FT AS A CLRNC; BUT NEITHER I NOR THE PF RECALL THAT; OR READ THAT BACK BEFORE TKOF. PRIOR TO THIS; A CITATION TOOK OFF BEFORE US WITH CLRNC OF A R TURN TO 100 DEGS. THEY WENT L AND TWR WAS FRANTICALLY TRYING TO GET THEM TO TURN R. WE BELIEVE THAT WAS THE ACFT THAT CAME CLOSE TO US CIRCLING BACK IN THE DIRECTION THEY WERE SUPPOSED TO GO. ALSO; THERE SEEMED TO BE SOME CONFUSION ON THE TWR'S PART ON WHICH ACFT HE WAS TALKING TO. WE THINK TWR WAS TRYING TO TELL THE ACFT THAT CAME CLOSE TO US TO STAY AT 2000 FT BUT USING THE WRONG TAIL NUMBER AS AN ACFT WAITING TO DEPART AFTER US SAID; 'WE ARE ON THE GND YET; YOU MUST MEAN THE OTHER ACFT.' WE BOTH BELIEVE THAT WE FLEW THE CLRNC GIVEN US.