This is one of those problems that so hard to solve. Silicone, latex... I've tried them all, and all eventually allow mildew and mold to grow. Caulk needs to be redone regularly (like every 10 years, no matter what the tube says)... it may seem gross, but it's not going to harm you.
So to truly get rid of it (for awhile), you need to remove all the old caulk that you can. The black mold "around" the caulk is probably mold working its way in from the edges.
So the first step is scraping that caulk out as completely as you can, and the trick is that you have to use tools that won't hurt the tile.
There is a magic tool out there, and you can usually get them for free. The tool is Formica samples (or any laminated counter-top material). This is one of the greatest unknown scraping tools... it's like a fingernail on steroids. And it can be shaped and sharpened with a mill bastard file as needed.
It's an extremely hard laminate of paper and adhesive under pressure... go to HD or Lowes and grab a few smooth (not textured) samples (they usually don't charge you, even when there's a sign saying 50¢ a sample or something).
Start working at the caulk edges with this... FINISH SIDE AGAINST THE TILE (not the brown side), and pull as much out as you can in full strips. You may be surprised at how easily it comes out, especially if it's a decade old or more. The goal is to reveal the gap that the original caulk tried to fill.
If you're as obsessive as I, you can use an abrasive pad for the remnants, perhaps with the hand cleaner Gojo (magic stuff... it has a grit that really helps... it's an automotive hand cleaner, but we use it for a lot of things). Another of my favorite items for tasks like this is copper wool (not steel wool – that would likely damage the tile surface, and you DON'T want that. Copper is a much softer metal than steel.
After you've removed all the old caulk that you can, it's time to sterilize the area as much as you can (an impossible task unless they used epoxy grout).
So a lot of people use bleach... I've already suggested Gojo and copper wool or a brass brush, but there is something more effective than bleach... it's called RMR-86 Pro, available on Amazon (I have no connection to the company whatsoever, but if you can't tell yet that I'm particular about stuff then so be it). It really is amazing, and it really is probably toxic... mainly to your eyes, so use this with Goggles and a good mask (or hold your breath and squint like we do, but we're old ...) It's the best final step.
Oh... bonus suggestion... the best shower glass cleaner by far is Cerama Bryte... the ceramic stove-top cleaner. Cuts through soap scum and water stains like magic.
Hope that helps... and when I've said "we" I mean myself and amazing wife.
