Originally from Buncrana in County Donegal, Mr Bradley studied for the priesthood in Rome and found himself serving in the Long Tower parish just as the Provisional IRA's campaign of violence was taking off.
"Suddenly I was in the middle of violence - and it was vicious", he told the Red Lines podcast.
"And what I discovered mostly was, violence is like addiction - it's inclined to get worse".
Four or five deaths in a short period of time
Mr Bradley, who later went on to become a film-maker and to co-chair the Consultative Group on the Past alongside Lord Eames, reflected at length on the challenges he faced on the streets of Derry five decades ago.
"Once you come up against the child dying on the street or the young British soldier dying in front of you... it brings you to a place that you don't often get to in life", he said.
"Within the parish that I worked... there were four or five deaths within a very short period of time. Two of them happened to be IRA men. One of them happened to be a man who I had got to know a bit and was really beginning to like... a beautiful man who was shot by a trigger-happy British soldier who was probably nervous about where he'd been left.
"(There was also) a young British soldier who was blown to bits. I ended up dealing with his body - but then also dealing with the person who blew him up within the hour of it happening.
"The young man who was shot by the IRA was an informer. All of that happened within a couple of hundred yards... within the parish."
'I still cry when I talk about it'
Fifty years on, those experiences still take their toll.
"I still cry when I talk about it," he told the podcast.
"It moulds me. It makes me who I am. You don't put it in a box and put it away, otherwise that just becomes a sickness," he added.
He was so affected by what he witnessed on the streets of Derry that he went on to become a secret "backchannel" between the IRA and the government, something he also reflected on in his recently published memoirs.
His involvement in the politically charged policing and legacy debates can also be traced back to his time in the priesthood and his first-hand experiences of violence.
"That doesn't destroy you - it humanises you," he said.
"It makes you aware of the need for talk and aware of the need for debate and aware of the need for politics and aware of the need for compassion and for empathy and decency and you recognise it when you see it in others."