DURING CLBOUT A C5 CREW HAVE AN NMAC WITH A VFR BEECH KING AIR OVER SCK.

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WE WERE ON LEG 2 OF A HAWAII-CA-EAST COAST TRIP; SO HAD BEEN UP FOR AT LEAST 14 HRS. ON CLBOUT WE PASSED THROUGH A THICK LAYER OF CLOUDS INTO A THINNER LAYER WITH NO DISCERNABLE HORIZON. DEP SWITCHED US TO CTR; CLBING THROUGH 11000 FT MSL. UPON CHKING IN CTR ADVISED US OF VFR TFC AT 1-2 O'CLOCK POS; ABOVE US. MY REPLY WAS 'WE'RE IMC;' AND MY REMARK TO THE CREW WAS; 'WHAT IS HE DOING FLYING VFR IN IMC?' ATC THEN ADVISED US TO 'INCREASE CLB THROUGH 16000 FT FOR TFC.' HE ASKED THE VFR TFC IF HE HAD US IN SIGHT; WHICH HE DIDN'T. THE TFC THEN APPEARED ON TCASII AS A TA AT 1 O'CLOCK POS; 7-8 MI +500 FT LEVEL. ATC HAD ASKED HIM TO DSND; SO I OBSERVED A DOWN ARROW ON THE TARGET. BEFORE THIS; WHEN I INITIALLY GOT THE TA I ASKED THE COPLT TO LEVEL OFF AT 13000 FT. I DID NOT WANT TO CLB THROUGH A VFR TFC ALT AT 13500 FT SINCE HE HAD NO VISUAL CONTACT WITH US. WHEN I NOTICED HIM DSNDING ON TCASII I DIRECTED THE COPLT TO TURN R. AFTER HE ROLLED TO 40 DEGS OF BANK; I RECEIVED AN RA 'MONITOR VERT SPD;' AND RED FROM ZERO FPM AND BELOW. THE COPLT HAD STARTED A SLIGHT DSCNT IN THE TURN SO I APPLIED SLIGHT BACK PRESSURE TO 'GET THE NEEDLE OUT OF THE RED.' I WAS BACK AND FORTH INSIDE AND OUT LOOKING FOR TFC; AND WATCHING MY TCASII SCOPE. THE RED DOT KEPT GETTING CLOSER TO THE ACFT SYMBOL. ABOUT 10 SECONDS AFTER THE INITIAL RA; THE TCASII CHANGED THE RA TO 'CLB; CLB NOW' AND I NOTICED THE RED RESTR GO TO 1000 FPM CLB; AND I NOTICED THE RED RA TFC DOT TOUCHING THE NOSE OF MY TCASII REF AIRPLANE SYMBOL. I APPLIED SMOOTH BUT SURE BACK PRESSURE TO PULL THE NEEDLE UP (VVI) OUT OF THE RED. DURING THE PULL THE AIRPLANE ENTERED AN ACCELERATED STALL AT APPROX 270 KTS AND 40 DEG OF BANK INDICATED BY A BUFFET FELT TO MOVE FORWARD UP THE FUSELAGE -- CULMINATING IN A STALL WARNING/STICK SHAKER FOR THE FINAL 2 SECONDS. THE TCASII RA THEN RESOLVED AND WE STARTED A SIGNIFICANT CLB; SINCE UPON THE CLB RA THE THROTTLES WERE PUSHED UP TOO. WE NOTIFIED ATC OF AN RA AND THAT I THOUGHT IT WAS VERY CLOSE. MY FE THEN NOTIFIED ME THAT THE AIRPLANE HAD EXCEEDED ITS G-LIMITS OF 2.33; AND THAT WE HAD TO RTB (RETURN TO BASE) TO INSPECT THE AIRPLANE. DURING THIS I NEVER SAW THE CONFLICTING ACFT AND WAS VERY UNEASY ABOUT HOW CLOSE HE CAME. I FELT I COULD NOT HAVE OUT-CLBED HIM WHEN ATC ASKED FOR IT; AND I RECEIVED THE INITIAL TA. I REALIZE THE PROC IS TO ROLL LEVEL THEN COMPLY WITH THE RA. MY POST EXPERIENCE I GUESS DROVE ME TO ROLL AWAY FROM THE TFC. ATC INFORMED ME ON THE GND THE RADAR PLOT SHOWED 400 FT AND 2.3 MI DIFFERENCE. I FELT IT WAS MUCH; MUCH CLOSER FROM THE DATA THAT APPEARED ON MY TCASII SCOPE. I FEEL THERE WERE NUMEROUS BROKEN LINKS IN THE CHAIN OF EVENTS THAT LED UP TO THIS CONFLICT. THE CTLR HAD ASKED THE KING AIR IF HE HAD US IN SIGHT A FEW TIMES BEFORE AND AFTER. THE KING AIR PLT INDICATED HE HAD TRIED TO GET AN IFR CLRNC FROM THE PREVIOUS CTLR; BUT ATC WAS TOO BUSY AND HANDED HIM OFF TO THE NEXT CTLR; WHO WAS ALSO VERY BUSY. THE WX WAS VERY HARD TO DISCERN VMC OR IMC FOR LIGHTNING; CLOUD LAYERS; ETC. WE WEREN'T IN THE CLOUDS; BUT I FEEL WE SURE WEREN'T VMC. THE CTLR'S WORKLOAD WAS HIGH AND HE PROBABLY DIDN'T HAVE MUCH TIME TO REACT. I HAVE HAD SOME RA'S BEFORE; BUT HAVE ALWAYS SEEN THE CONFLICTING ACFT BEFORE IT GOT TOO CLOSE. THIS IS THE FIRST TIME AND ONLY TIME I HAVE NEVER SEEN THE TFC. QUITE DISCONCERTING AND FRANKLY VERY SCARY. I WAS VISIBLY SHAKEN FOR ABOUT 15-30 MINS AFTERWARDS AND FOUND IT HARD TO CONCENTRATE ON MY JOB OF 'TALKING ON THE RADIOS.' I KEPT TAKING MYSELF BACK TO THE SIT TO TRY TO RECTIFY OR EXPLAIN TO MYSELF WHAT I COULD HAVE BEEN DONE DIFFERENTLY. I STILL; TO THIS DAY; REFLECT UPON THE SIT IN A SIMILAR MANNER. I WAS SURPRISED HOW EASY IT WAS TO PULL THE ACFT INTO A STALL. MY AIRLINE TRAINING TAUGHT ME TO RESPECT THE RA BUT DON'T ASK TOO MUCH OF THE AIRPLANE. I REALLY DIDN'T THINK I PULLED VERY HARD ON THE YOKE. I WAS WHAT I FEEL A CLASSIC ACCELERATED STALL; STARTING AT THE WINGTIPS; THEN MOVING FORWARD. ANOTHER THING OF WHAT HAPPENED; RIGHT AFTER; WAS THE COPLT ENTERED A L BANK; ALMOST AN UNUSUAL ATTITUDE. I WASN'T SURE IF HE WAS FLUSTERED LIKE MYSELF OR IF THE ACFT HAD DEPARTED CTLED FLT. I OFTEN WONDER IF I HADN'T HAD TCASII; WOULD I HAVE BEEN SO SHAKEN? WOULD I HAVE MISSED HIM? I DEFINITELY WOULD HAVE TRUSTED ATC MUCH MORE. FATIGUE ALSO WAS A FACTOR. WE HAD BEEN UP FOR OVER 14 HRS AFTER A VERY EARLY (LCL TIME) MORNING WAKE UP IN HAWAII. SO GOING ON 4-5 HRS OF SLEEP; PLUS A SMALL CAT NAP ON THE AIRPLANE. I WONDER IF OUR RESPONSE WOULD HAVE DIFFERED; IF THIS HAPPENED LEG 1 OF THE DAY. I THINK THE KING AIR WAS PUSHING TO GET TO HIS DEST. RATHER THAN CIRCLE AND MAINTAIN VFR; HE KEPT ON; AND THE WX KEPT GETTING WORSE. I DON'T FAULT THE CTLR PERSONALLY. I'M SURE HE DID HIS BEST; GIVEN THE TOOLS HE HAD TO WORK WITH; BOY THERE WAS A LOT OF TFC AND THE CTLR WAS WHAT I PERCEIVE AS 'OVERLOADED.'