IFR BE20 DEP FROM RYY EXPERIENCED NMAC WITH VFR TFC ENTERING PATTERN.

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RWY WAS VERY BUSY WITH ACFT BEING TIGHTLY SEQUENCED (SEVERAL ACFT MAY HAVE BEEN PLTED BY INEXPERIENCED OR STUDENT PLTS). WX WAS CAVU; AND APPARENTLY MANY PLTS WERE OUT FLYING ON THIS SATURDAY AFTERNOON. OUR ACFT; A KING AIR 200; WAS CLRED INTO POS AND HOLD ON RWY 9; WE WERE ON AN IFR FLT PLAN. OUR CLRNC AFTER DEP WAS THE FOLLOWING: RADAR VECTORS TO HARRIS VOR; CLB TO 3000 FT MSL; HDG 360 DEGS AFTER DEP. AN ACFT (PIPER CUB OR TRI-PACER TYPE) WHICH HAD JUST LANDED WAS SLOW CLRING THE RWY; AND WE WERE DELAYED. AS WE WERE CLRED FOR TKOF; ANOTHER ACFT ON APCH WAS INSTRUCTED TO GO AROUND. (IT IS INTERESTING TO NOTE THAT WE HAD EXPERIENCED THE SAME SIT WHEN WE HAD PREVIOUSLY LANDED -- WE HAD TO INITIATE A GAR BECAUSE TFC HAD NOT BEEN SEQUENCED PROPERLY.) AS WE CLBED OUT AND BEGAN OUR TURN TOWARD A 360 DEG HDG; WE WERE TOLD TO CONTACT DEP. AS WE APCHED 2500 FT MSL; AN ACFT (IT APPEARED TO BE A C400 SERIES ACFT) MOVED IN FRONT OF US FROM R TO L. WE IMMEDIATELY STOPPED THE TURN AND DEVIATED TO THE R. THE CONFLICTED ACFT APPEARED TO BE AT THE SAME ALT AND ENTERING A L DOWNWIND TO RWY 9. WHEN WE CONTACTED DEP; WE WERE TOLD TO WATCH FOR TFC AT 2500 FT MSL AT 2 O'CLOCK POS. HOWEVER; BY THAT TIME THE ACFT WAS PAST US TO THE L (THE W). WE SUBSEQUENTLY ASKED CTLR ABOUT THE CONFLICT. HE REPLIED THAT THE ACFT HAD BEEN HANDED OVER TO RYY TWR AND THAT HE WAS NOT IN CONTACT WITH THE ACFT. THIS NEAR MISS APPARENTLY OCCURRED AS CTL FOR BOTH ACFT WAS BEING PASSED FROM APCH TO TWR AND TWR TO DEP. THE BUSY TFC ENVIRONMENT AT RYY AND OUR DELAY ON THE GND WERE CONTRIBUTING FACTORS SINCE THEY MAY HAVE CAUSED BOTH ACFT TO BE OUT-OF-SEQUENCE FROM WHAT WAS PLANNED BY ATC.