Joey's fame came from three main areas:
(1) His enormous talent as a comic - quick, thinking on his feet, going for the thrill or "kick" of a brilliant turn-around.
(2) His membership in the Sinatra-centric "Rat Pack" - again, the mix of talent and Hell raising were the key.
(3) His two television shows which, despite their short lifespans, made him part of the early and mid-1960s TV culture (and introduced Regis Philbin to the world).
Known as "The Frown Prince" because of his characteristic taciturn expression, it isn't a surprise that Bishop was born with the Sun in Capricorn - a Sun-sign known for some of the greatest comics, some of the most dedicated Hell raising, and of course a Mars-Saturn temperament and presentation. Sun was also closely opposite Saturn, a hallmark aspect for comics. Bishop wore the Mars side of Capricorn quite visibly, too - contrary to most late night talk show hosts, he had a challenging and sometimes almost attacking style barely held in rein. If you know Joey and his work from way back, read the Sun in Capricorn traits http://solunars.net/viewtopic.php?f=16&t=7#p24 and you'll see him all over the list.
He wasn't just a Capricorn - a Rim type - he was a double Rim, having also a Libra Moon. This and a close Mars-Pluto square establish his "Rat Pack" spirit, building on the prankish juvenescence, Pan-like spirit of the Sign of the Goat.
His humor delivery was very much in the style of Mercury-Mars. Since the two are not in aspect, it's pleasing to see the strongly-aspected Mars in Virgo.
At the time of his death, Neptune had been transiting sesquisquare natal Mars. Transiting Mars had conjoined his Pluto and squared his Mars a few days earlier - without more information on his final days, it's hard to say decisively what this meant, though it would be expected to be a hard kick to a failing body. Mercury transited in square to natal Neptune roughly the hour of his death (so I wouldn't be surprised to learn that respiratory failure was basic to the final stage). That's all we can tell from transits without a timed birth.
Secondary Progressions show a close Mars-Jupiter square and the possibility that progressed Moon was near transiting Neptune. Progressed Sun had just entered the square to his natal Sun. Most of these would be expected to be heightened vitality and pro-health aspects, so it isn't easy to ready them absent specific details of his death and, more importantly, absent a birth time.