First lets make sure we know the scale of the solar system. Neptune, the most distant major planet, has an orbit of 165 years. Sedna, the object with the longest known period, has an orbit of 11400 years.
Black holes come in different sizes. A black hole that has formed from the collapse of a giant star would have a mass more than three times the mass of the sun. If a black hole of this size were inside the orbit of Neptune, it would not be orbiting the sun, the sun would be orbiting it.
How about further out? Blackholes can be spotted in X-rays when matter falls into them forming a disc of matter orbiting at very high speeds, and heating up to enormous temperatures. A black hole without an accretion disc would be very dark and hard to detect from a distance. But we can be reasonably confident that there are no in our immediate neighbourhood.
The motion of a blackhole in orbit with the sun would be described by Newtons laws of gravity. The sun and the black hole would orbit on elliptical orbits around their common centre of gravity. Planets of the sun or the black hole would be stable within their Hall radii. Close to the blackhole one would need General relativity to describe the gravitational field, but from further out, its gravity would be almost exactly as predicted by Newton. You could model it in any gravity simulator.
It is possible that micro black-holes exist in the universe. These could pass through the earth without doing much damage, as the are sub-atomic size and since they interact only with matter by gravity, and don't weigh much, they don't have much effect. They might be "hot" with Hawking radiation.
None of these are much of a risk to the survival of the solar system. Black holes are rare.
