Others (dmckee and can-ned food) have already pointed out that the story you're looking for is from Fred Saberhagen's Berserker series. There are many stories in this series. I believe you're asking about the ones where Johann Karlsen purposely gets himself and a pursuing berserker trapped in an orbit around a black hole. Karlsen is featured in the four stories described below.
1. "Stone Place": novelette, first published in If, March 1965, available at the Internet Archive. ISFDB synopsis:
The various human societies inside and outside the Solar System form an uneasy alliance to launch a massed attack on the mechanized Berserker fleet which is dedicated to destroying all life.
This story introduces the character Johann Karlsen, leader of the fleet of the human alliance in a great battle against the berserkers:
Throughout the long war, the berserker computers had gathered and collated all available data on the men who became leaders of Life. Now against this data they matched, point for point, every detail that could be learned about Johann Karlsen.
The behavior of these leading units often resisted analysis, as if some quality of the life-disease in them was forever beyond the reach of machines. These individuals used logic, but sometimes it seemed they were not bound by logic. The most dangerous life-units of all sometimes acted in ways that seemed to contradict the known supremacy of the laws of physics and chance, as if they could be minds possessed of true free will, instead of its illusion.
And Karlsen was one of these, supremely one of these. His fitting of the dangerous pattern became plainer with every new comparison.
2. "Masque of the Red Shift": novelette, first published in If, November 1965, available at the Internet Archive. ISFDB synopsis:
A robotic representative of a giant artificially intelligent war machine programmed to destroy all life infiltrates the decadent court of the human rulers.
The black hole, or "hypermass" as it's called, is introduced at the beginning of this story:
Nogara had not come here to look at galaxies, however; he had come to look at something new, at a phenomenon never before seen by men at such close range.
It was made visible to him by the apparent pinching-together of the galaxies beyond it, and by the clouds and streamers of dust cascading into it. The star that formed the center of the phenomenon was itself held beyond human sight by the strength of its own gravity. Its mass, perhaps a billion times that of Sol, so bent spacetime around itself that not a photon of light could escape it with a visible wavelength.
The dusty debris of deep space tumbled and churned, falling into the grip of the hypermass. The falling dust built up static charges until lightning turned it into luminescent thunderclouds, and the flicker of the vast lightning shifted into the red before it vanished, near the bottom of the gravitational hill. Probably not even a neutrino could escape this sun. And no ship would dare approach much closer than Nirvana now rode.
It ends with Karlsen and the pursuing berserker flying into the hypermass:
The launch was now going certainly into the hypermass, gripped by a gravity that could make any engines useless. And the berserker-ship was going headlong after the launch, caring for nothing but to make sure of Karlsen.
The two specks tinted red, and redder still, racing before an enormous falling cloud of dust as if flying into a planet's sunset sky. And then the red shift of the hypermass took them into invisibility, the the universe saw them no more.
3. "In the Temple of Mars": novelette, first published in If, April 1966, available at the Internet Archive. ISFDB synopsis:
A gladiator becomes a pawn in the schemes of a death cult which worships Berserker machines.
Karlsen is offstage in this story, but a plan for rescuing him is discussed:
"Yes, I think there is a chance." Hemphill's face had become iron again. "You saw what efforts the berserkers made to kill him. They feared him, in their iron guts, as they feared no one else. Though I never quite understood why . . . so, if we can save him, we must do so without delay. Do you agree?"
"Certainly, but how?"
"With this ship. It has the strongest engines ever built—trust Nogara to have seen to that, with his own safety in mind."
Mitch whistled softly. "Strong enough to match orbits with Karlsen and pull him out of there?"
"Yes, mathematically. Supposedly."
4. "The Face of the Deep": short story, originally published in If, September 1966, available at the Internet Archive. ISFDB synopsis:
A hero in the war against machines which intend to exterminate all life is rescued from orbit around a black hole.
This story opens with Karlsen, in orbit around the hypermass, admiring the view outside his craft:
After five minutes had gone by with no apparent change in his situation, Karlsen realized that he might be going to live for a while yet. And as soon as this happened, as soon as his mind dared open its eyes again, so to speak, he began to see the depths of space around him and what they held.
There followed a short time during which he seemed unable to move; a few minutes passed while he thought he might go mad.
He rode in a crystalline bubble of a launch about twelve feet in diameter. The fortunes of war had dropped him here, halfway down the steepest gravitational hill in the known universe.
At the unspeakable bottom of this hill lay a sun so massive that not a quantum of light could escape it with a visible wavelength. In less than a minute he and his raindrop of a boat had fallen here, some immeasurable distance out of normal space, trying to escape an enemy. Karlsen had spent that falling minute in prayer, achieving something like calm, considering himself already dead.
But after that minute he was suddenly no longer falling. He seemed to have entered an orbit—an orbit that no man had ever traveled before, amid sights no eyes had ever seen.
He rode above a thunderstorm at war with a sunset—a ceaseless, soundless turmoil of fantastic clouds that filled half the sky like a nearby planet. But this cloud-mass was immeasurably bigger than any planet, vaster even than most giant stars. Its core and its cause was a hypermassive sun a billion times the weight of Sol.
[. . . .]
His orbit, he guessed, must be roughly the size of Earth’s path around Sol. But judging by the way the surface of clouds was turning beneath him, he would complete a full circuit every fifteen minutes or so. This was madness, to
outspeed light in normal space — but then of course space was not normal here. It could not be. These insane orbiting threads of dust and rock suggested that here gravity had formed itself into lines of force, like magnetism.
The orbiting threads of debris above Karlsen’s traveled less rapidly than
his. In the nearer threads below him, he could distinguish individual rocks, passing him up like the teeth of a buzzsaw. His mind recoiled from those teeth, from the sheer grandeur of speed and distance and size.
I think you must have read more than one of these stories; you certainly must have read "The Face of the Deep", because that's the only one that depicts Karlsen in orbit around the black hole, admiring the scenery; but I think you must have read one or more of the earlier stories as well. All four of them appeared in the collection Berserkers: The Beginning which was published in 1998, agreeing with your "late 90s." Alternatively, you could have read them in in the earlier collection Berserker, perhaps in one of the Ace Books editions.
