This may be a really basic question with the simplest of answers, but as much sense as it makes to think that black holes are not, in fact, black but invisible - because light would be gravitationally lensed from all directions around the black hole, 'covering' the hole - I am yet to find an artist's impression portraying the black hole as anything other than, well, a 'black hole' in space. Even the movie 'Interstellar', which is meant to have unimaginably realistic graphics and has undergone a gigantic amount of physical simulations to achieve such, portrays the black hole in that stereotypical manner. Or, perhaps, my intuitive speculations are unjustified?
Light that is coming from a point behind a black hole "bends around" it so you can see it - but you don't see it as "appearing to come from the black hole". The effect is called "gravitational lensing". This diagram might make that clear (source):
The result is that light "from behind" the black hole will appear on the rim - like this image (source)
Sometimes the light from a single galaxy can appear as a streak or even an almost complete circle, depending on the exact position relative to the black hole and the observer. Lots of examples when you google "gravitational lensing".
In short - no light can appear to come "from" a black hole. The artists (and the physics engines from Interstellar) got it right.
If you mean invisible as in see-through transparent, then no, light does not pass through a black hole. Light is absorbed and cannot escape from a black hole. So you will see an absence of any photons coming from a black hole, totally black.
Gravity curves space and time and the gravity of a black hole is immense, so I would not trust the light that gets to us that encounters the near periphery of a black hole. That light would be bent and distorted from its original form when we observe it.
Gravitational lensing as discussed in the other answers will indicate whether a massive system is there. Radiation identifiable as coming from the black hole, so as to mask its "blackness", i.e total absorption of radiation, is seen here
This composite image shows powerful radio jets from the supermassive black hole at the center of a galaxy in the Phoenix Cluster inflating huge "bubbles" in the hot, ionized gas surrounding the galaxy. The cavities inside the blue region were imaged by NASA's Chandra X-ray observatory. Hugging the outside of these bubbles, ALMA discovered an unexpected trove of cold gas, the fuel for star formation (red). The background image is from the Hubble Space Telescope
Charged particles, attracted and falling into a black hole, will emit radiation, also particles circulating in the gravitational field of a black hole. Hawking radiation is not in the visible for large black holes so cannot be caught in astronomical measurements. Considering also that the black hole may have a charge or a magnetic field, it is inevitable that part of this radiation will mask the "blackness".
So black holes, as seen above, are calculated to exist behind a curtain of visible radiation, fitted with various models. In the image above the charged particles form plasma jets.
With some patience , we might get a better resolution of how a "physical black hole" looks, in contrast to an artist's image:
After training a network of telescopes stretching from Hawaii to Antarctica to Spain at the heart of our galaxy for five nights running, astronomers said Wednesday they may have snapped the first-ever picture of a black hole.
It will take months to develop the image,...
Already we see that in real life photographs by telescopes of black holes, the blackness is masked . The artists' impressions are really two dimensional cuts in a three dimensional space, which is not something observable without mathematically unfolding data.
a black hole is neither black nor invisible as it is an absence of matter, photons or anything you can possibly think of (if it helps think of it as a vacuum within a vacuum) it just looks black to us due to the absence of reflected light as anything black absorbs all light, if a black hole was invisible then light would pass through it and we would be unable to identify it was there until it was too late.