I used to have one of those Pur filters. Works great. My water wasn't as gross as yours seems to be though. You might have to replace the filter more frequently. Also, that green light you see runs off a battery that you can't replace. Unless they changed it in the last few years.
I also poured the filtered water through a pitcher filter just to be extra.
You can also consider getting a Reverse Osmosis water filtration system. Hooks up to the cold water line and hidden under the sink with a faucet running to the top. Will run you a few hundred more or less depending on brand and features. Good clean drinking water is worth it though.