Do you mean the reaction with Cl2 or with Cl- ions? In either case, I don't think that there is a positive charge on the chlorine attached to the boron.
With respect to "who attacks who" by convention electrons always do the attacking, so it would be the electrons from chlorine (or chloride) that attack the vacant p orbital on boron to form the tetrahedral complex. The attack can come from either side because BF3 is trigonal planar. The product is probably a covalently bound complex, so sharing of electrons. The electronegativity difference between B and Cl is boderline for ionic vs. covalent, but the B is already electron deficient from the 3 Fs.