Would a black hole passing next to a star create a deadly focal point due to gravitational lensing?
tl;dr: seen at a distance, the lensed object will appear as an annulus or &quot;ring&quot; around the lensing object, and while that will be brighter than if there were only empty space, sadly it won't be death-ray bright!

Let's first think about what makes a familliar lens a lens. Near the center the thickness doesn't vary much, but as you move farther from the center the thickness changes faster and faster.
If we measured the slope or angle of the surface, we'd see that the angle increased roughly linearly with distance from the center.
In the thin lens approximation the angle that the lens bends the light $\Delta \theta$ is proportional to the distance from the center of the lens $r$ where the light hits it. The bending power increases linearly with distance.
$$\Delta \theta \approx \frac{r}{f}$$
where $f$ is the focal length of the lens.
How do concentrated points of mass bend light?  Wikipedia's Gravitational lens give us
$$\Delta \theta \approx \frac{4 G M}{r c^2}$$
and that's a problem because now $r$ is on the bottom!
Single concentrated objects like black holes do not act like the familliar lenses we use to focus. They do have some ability to concentrate somewhat compared to empty space, but nowhere near as much as real lenses do. From a source that's a certain distance away, there will be only one angle that gets bent parallel into a &quot;beam&quot;, angles slightly larger or smaller passing slightly farther or closer from it will be bent much less or much more, either diverging or converging from the axis afterward.
Thus seen at a distance, the lensed object will appear as an annulus or &quot;ring&quot; around the lensing object, and while that will be brighter than if there were only empty space, sadly it won't be death-ray bright!
In the case of your star/BH pair, exactly how bright it will be along that axis depends on the details, but one can get a good estimate by ray tracing, either with a few lines of Python or pencil and paper for estimate purposes.
If the star were close then it is an extended object, a wide disk, and those can't be focused to points even by lenses (we can't concentrate blue sky with a magnifying glass) so there wouldn't be much of a death effect. If it were far, you could concentrate it better (because it's more point-like) but it would be a lot dimmer to begin with because it would be farther away.

A Horseshoe Einstein Ring from Hubble  The bright thing in the middle is the lensing mass, the ring is the lensed object behind it. It's not a &quot;beam&quot; but just a distorted view, but it is brighter than if the lens weren't there.