ACFT ENTERS ACTIVE RWY WITH ACFT ON FINAL AT PAO.

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WHILE DEPARTING RWY 30 AT PAO ON AN IFR FLT PLAN; I HEARD A PLANE ANNOUNCE THAT HE WAS 'GOING AROUND.' THE ANNOUNCEMENT WAS HEARD AFTER APPLYING TKOF PWR AND AFTER A SAFE STOP COULD HAVE BEEN MADE. AFTER ROTATING (DURING CLBOUT); I HEARD THE PLT IN THE CESSNA STATE THAT HE CALLED FINAL AS I ENTERED THE RWY. BEFORE ENTERING THE RWY; I WAS ON BAY DEP FREQ 121.3; ATTEMPTING TO GET MY IFR CLRNC. WHEN UNABLE TO ESTABLISH CONTACT ON FREQ; I USED A CELL PHONE TO CONTACT BAY AND RETRIEVE THE CLRNC. IMMEDIATELY AFTER RECEIVING MY CLRNC; I PERFORMED A FINAL TKOF CHKLIST; SWITCHED TO ADVISORY FREQ AT 118.60 AND ANNOUNCED MY DEP. SINCE I WAS RUSHING TO DEPART (THE CLRNC HAS A TIME TO DEPART); I FAILED TO VISUALLY CHK FINAL APCH FOR INCOMING TFC. NOR DID I ASK FOR A REPEAT OF A GARBLED XMISSION THAT OCCURRED AFTER MY DEP XMISSION. I NEVER MADE VISUAL CONTACT WITH THE CESSNA; EVEN AFTER TKOF. I THINK CONTRIBUTING FACTORS WERE: 1) LACK OF A GOOD NIGHT'S REST THE PREVIOUS NIGHT. 2) LACK OF SOUND JUDGEMENT ON DETERMINATION OF PLTING CAPABILITY DUE TO LACK OF SLEEP AND WORKING A 13 HR DAY. 3) UNDETERMINABLE CEILING -- I THOUGHT THAT THE CEILING COULD HAVE BEEN AS LOW AS 300 FT OVCST; WHICH I ASSUMED; AND WHICH MADE ME THINK VFR TFC COULD NOT BE FLYING; AND IFR TFC WOULD NOT BE INBOUND SINCE I HAD A CLRNC TO DEPART. 4) LACK OF TUNING THE RADIO TO ADVISORY FREQ DURING THE FINAL TKOF CHKLIST. 5) LACK OF VISUAL CHK OF FINAL APCH. 6) TRYING TO HURRY THE TKOF. 7) FOCUSING ON THE DEP/IFR CLRNC INSTEAD OF THE TKOF. CORRECTIVE ACTION WAS TAKEN BY THE CESSNA BY GOING AROUND. IN THE FUTURE; I PLAN TO FIND OR DEVELOP A FATIGUE SELF-TEST THAT I CAN ADMINISTER TO MYSELF TO DETERMINE IF I AM TOO FATIGUED TO FLY.