IPhone case with a chain

How would that lanyard connect to a normal case ?

I would look for one with an end like the one below.  if you can put a small hole in the case, feed the lanyard loop through and put the case back through the hole securing it.   There are several styles and sizes with different ends.  

Here is another example, you just want to make sure the "loop" on the end will open large enough to slip your phone case back through the end.  

How would that lanyard connect to a normal case ?

It won't. The case needs to be specially adapted to take a cord of any kind.

Depends what your idea of a normal phone case it.

That case shown is adapted to a cord. None of mine ever have been.

No, I simply looped it through an existing opening in the case.  If the case didn't have a small hole, I could have drilled one.