F900 CREW RECEIVED 2 TCASII RA'S DEPARTING RDU.

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ACFT ON DEP CLB FROM RWY 23L RDU ON RWY HEADING CLBING TO 7000 FT MSL. AT APPROX 5500 FT MSL TCASII RA SOUNDED. CREW DETERMINED THE ALERT WAS BASED ON RATE OF CLB AND WAS NOT AN ACTUAL CONFLICT. DURING THIS RESOLUTION ACTIVITY; RDU DEP CTL ISSUED CLRNC TO TURN R TO 330 DEGS. SHORTLY AFTER INITIATING THE R TURN; A SECOND RA SOUNDED. AT THIS TIME; THE ACFT WAS CLBING THROUGH 6000 FT MSL. IN THE PROCESS OF RESPONDING TO THE RESOLUTION THE ACFT EXCEEDED THE CLRED ALT BY 200-300 FT. THE ACFT LEVELED AT THE CLRNC ALT WHEN THE RESOLUTION CLRED. DURING BOTH RESOLUTIONS; THE SIT WAS COMPOUNDED BY THE VOLUME OF THE TCASII VOICE ADVISORY WHICH OVERRODE COM WITH ATC AND BTWN FLC MEMBERS. VISUAL CONFIRMATION OF THE TCASII ALERT WAS NOT POSSIBLE DUE TO HIGH VOLUME OF TFC AT THE HIGH DENSITY ARPT. SUPPLEMENTAL INFO FROM ACN 488144: DEP GAVE US A R TURN TO 330 DEG HDG. DURING TURN ANOTHER ACFT AT OUR 2 O'CLOCK POS; CAUSED THE TCASII TO ALERT OF RA. AT THIS POINT WE DID NOT KNOW WHICH TCASII RESOLUTION WE WERE TRYING TO AVOID.