Both circulation and vorticity have to do with rotation of a fluid element.How are vorticity and circulation are related.
No. If no vorticity then, no circulation.
Mathematically, The flux of vorticity is circulation.
$ \Gamma = \int\int \omega~ ds$
So, $\Gamma \rightarrow $ 0 when $\omega \rightarrow $ 0.
Physically, vorticity is just not a rotation of a fluid element. But it is the rotation of that element about its own axis (spin). (Please know about free/forced vortex for a kind of imagination).
So, circulation is non-zero only when there is a flux of vorticity.
The Biot-Savart Law can be interpreted as the relationship between the velocity induced by a vortex tube and the strength of the vortex tube, i.e. its circulation.
Note: you need to be a little careful in ascribing causes and effects due to the unfortunate adjective "induced" in induced velocity. In classical fluid mechanics there is no action-at-a-distance, so vorticity cannot cause changes to fluid velocities at remote locations.