Diplomatosi nel collegio reale militare canadese, scelse di operare tra i fucilieri canadesi. Si aggregò alle truppe inglesi dall'agosto 1914, ma presto, nel luglio 1915, fece domanda per diventare pilota d'aerei.
Iniziò come ufficiale osservatore nel 21º Squadrone e, alla fine dell'autunno1915, riportò delle gravi ferite in un incidente aereo. Bishop imparò poi a pilotare e, nel marzo 1917, si aggregò al 60º Squadrone per combattere al fronte. Il 25 marzo 1917 abbatté il suo primo aereo nemico; cinque mesi più tardi, alla fine di agosto, aveva già raggiunto 47 vittorie confermate. Nella primavera del 1918, quando comandò l'85º Squadrone in Francia, ottenendo altre 25 vittorie nello spazio di 12 giorni.
A Bishop venne assegnata inoltre la Victoria Cross, la prima conferita a un aviatore canadese, il 2 giugno 1917.
Morì l'11 settembre 1956 nel sonno. Il motivo è sconosciuto.
«For most conspicuous bravery, determination, and skill. Captain Bishop, who had been sent out to work independently, flew first of all to an enemy aerodrome; finding no machines about, he flew on to another aerodrome about three miles southeast, which was at least 12 miles the other side of the line. Seven machines, some with their engines running, were on the ground. He attacked these from about fifty feet, and a mechanic, who was starting one of the engines, was seen to fall. One of the machines got off the ground, but at a height of 60 feet, Captain Bishop fired 15 rounds into it at very close range, and it crashed to the ground. A second machine got off the ground, into which he fired 30 rounds at 150 yards range, and it fell into a tree. Two more machines then rose from the aerodrome. One of these he engaged at a height of 1,000 feet, emptying the rest of his drum of ammunition. This machine crashed 300 yards from the aerodrome, after which Captain Bishop emptied a whole drum into the fourth hostile machine, and then flew back to his station. Four hostile scouts were about 1,250 feet above him for about a mile of his return journey, but they would not attack. His machine was very badly shot about by machine gun fire from the ground.»
— 2 giugno 1917
«For conspicuous gallantry and devotion to duty when engaging hostile aircraft. His consistent dash and great fearlessness have set a magnificent example to the pilots of his squadron. He has destroyed no less than 45 hostile machines within the past 5 months, frequently attacking enemy formations single-handed, and on all occasions displaying a fighting spirit and determination to get to close quarter with his opponents which have earned the admiration of all in contact with him.»
«A most successful and fearless fighter in the air, whose acts of outstanding bravery have already been recognised by the awards of the Victoria Cross, Distinguished Service Order, Bar to the Distinguished Service Order, and Military Cross. For the award of the Distinguished Flying Cross now conferred upon him he has rendered signally valuable services in personally destroying twenty-five enemy machines in twelve days—five of which he destroyed on the last day of his service at the front. The total number of machines destroyed by this distinguished officer is seventy-two, and his value as a moral factor to the Royal Air Force cannot be over-estimated.»