δίκη, in Ancient Greek, was "law, justice". Nowadays, it's a "trial", i.e., where judgment is passed.
From that was derived the verb δικάζω "to judge, to render justice."
And from that the actor noun δικαστής "a judge, someone who renders justice".
I imagine that δικηγόρος is also related to this "law" root but I'm not sure where the ending comes from.
The delta at the beginning is supposed to sound like "th" as in English "this, that, though".
That sound and the "v" sound can sound similar as they both involve the teeth -- one is labiodental (lip against teeth = v), the other dental (tongue against/between teeth = th) -- and are both voiced.
But they're different.