Short of installing a reverse-osmosis (RO) filtration system under your sink, which you probably aren't allowed to do in an apartment, a countertop filter that just screws into the aerator on your faucet -- like this or this or this or this or this -- will do a better job than any Brita jug or tiny faucet-mount filter, without having to modify your under-sink plumbing. The filter cartridges are also industry-standard, cheaper and longer-lasting with a variety of filtration types available (you'll prolly want the activated charcoal type of filter for flavor improvement).
A countertop filter that just screws into the aerator on your faucet -- like this or this or this or this or this -- will do a better job than any Brita jug or tiny faucet-mount filter, without having to modify your apartment's under-sink plumbing. The filter cartridges are also a cheaper and longer-lasting industry-standard with a variety of filtration types available (you'll prolly want the activated charcoal type of filter for flavor improvement).