AVR146 DEPARTING AZO EXPERIENCED CONFLICT WITH VFR TFC.

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I WAS THE CAPT; PF. A PA28 HAD DEPARTED AZO AHEAD OF US AND WAS INSTRUCTED TO CLB TO 2500 FT. AZO TWR CLRED US FOR TKOF WITH A CLB TO 3000 FT ON RWY HEADING (RWY 17). WE HAD THE PA28 IN SIGHT FOR DEP. THE ACFT WAS ON A SLIGHTLY DIVERGING HEADING FROM US TO THE SW. AS WE APCHED FROM BEHIND AND TO THE L; WE WERE HANDED TO DEP CTL; WHO WAS NOT TALKING TO THE ACFT AHEAD. AS WE CROSSED THROUGH HIS ALT (2500 FT); WE WERE ABOUT 3/4 MI OFF HIS L AND SLIGHTLY BEHIND; ABOUT HIS 8 O'CLOCK POS. AT THIS TIME; HE TURNED INTO US. WE GOT A TA WHICH IMMEDIATELY TURNED TO AN RA; TO CLB IN EXCESS OF 4000 FPM. THE AUTOTHROTTLES WERE ENGAGED AND BEGINNING TO RETARD TO LEVEL AT 3000 FT. I TURNED THEM OFF AND APPLIED MAX CLB PWR AND PITCHED UP TO CLB. AS WE WERE GOING THROUGH 3000 FT; ATC CLRED US HIGHER WITHOUT MENTION OF THE TFC. THE TFC PASSED BELOW US NO MORE THAN 500 FT BELOW. WE CONTINUED OUR CLB TO HIGHER AND RECONFIGURED FOR A NORMAL CLB AGAIN. THE TFC CHKED IN WITH DEP AND SEEMED OBLIVIOUS TO OUR CLOSE ENCOUNTER. I CALLED AZO ATC AND QUESTIONED THEM ABOUT THE EVENT. THE SUPVR SAID THAT THE TFC HAD US IN SIGHT AND WAS INSTRUCTED TO MAINTAIN VISUAL SEPARATION WITH US. WE WOULD HAVE PASSED CLR OF HIM WITHOUT INCIDENT HAD HE NOT TURNED UNEXPECTEDLY. ATC DID NOT EXPECT THE PA28 TO TURN INTO US EITHER. AFTER REVIEWING THE TAPES; THE SUPVR SUGGESTED I FILE AN NMAC FORM. THIS COULD HAVE BEEN AVOIDED BY SEVERAL MEANS. ONE WOULD BE FOR ATC NOT TO CLR US TO GO SO SOON AFTER SLOWER TFC IN THE SAME GENERAL DIRECTION. WE SHOULDN'T ACCEPT THE CLRNC. WE COULD DELAY OUR ACCELERATION SEGMENT UNTIL CLBING CLR OF THE TFC (WE HAD ACCELERATED TO 250 KTS WHEN THIS OCCURRED).