I'm an American that has been living in Cologne for over 12 years. I do understand how many find the city underwhelming as a tourist. But the motto here is 'Cologne is a feeling'. She isn't the prettiest, but the people, and the history, make the city unique compared to other cities in Germany.
All the guide books can show you the main sites; of which there aren't many at the moment. Unfortunately, the city is in the middle of building a Roman History Walk through the center of Cologne. As a result, there is a lot of construction and many of the Roman museums/sites that make Cologne famous are closed, including the Roman Germany Museum right by the Cathedral. The 'replacement' museum they've set up across town is not worth the 6euros.
Definitely visit the Dom, and either read up about it or take a tour. The Dom Forum organizes tours of the cathedral, some in English. Go to the website, or walk in on the day (although English tours might be booked up, depending on the time of year). They also offer a rooftop tour that is really special. This one sells out very early and needs to be booked in advance, but you get to walk along to top of the Cathedral and see the workshop up there.
If you are here at Christmas market time, or at Karneval, there is nothing else you need to do but these things.
With 5 days you have plenty of time for day trips. Not sure if you're headed south into the wine country at some point, but I always take my visitors down along the Rhine. This is the best representation of what visitors have in their head when they think of Germany. During the high season the KB boats run a sort of hop on hop off service, so you can decide where to go. We always take a train from Cologne main station to Boppard. Boppard is very small, but cute, and the KD boat stops here. The Rhine doesn't get really pretty until after Koblenz, so take a train to save time. From Boppard take the boat down the Rhine and get off where you like. I like Bacarach (small but charming) or Rüdescheim (bigger and many more cruise ship tourists, but for a reason). If you've spent the day getting to Rüdescheim, have dinner and then take the train back to Cologne. It's a long day, but doable and worth it! If you have the time, spend a night someone along the Rhine. Warning, the smaller towns are totally dead after 10p.
After that, spend time in the famous breweries in Cologne. My favorite is Gaffel am Dom, right by the train station; you'll probably need a reservation which can be made online. People are quite friendly here and it's easy to strike up a conversation in a beer hall. Drink Kölsch and eat the hearty German food.
Cologne loves to party. Depending on when you're here, there are festivals almost all the time over summer. Find an event calendar and see what is happening when you're here.
If you are a history person (I'm a Roman nerd), it is possible to see Roman ruins throughout the city. Cologne was the northernmost capital of the empire and there are always ruins being discovered, even now. It's possible also to walk the Roman wall that once rounded the city; but you need a guidebook. After the Romans, the city expanded further; you can also see some of the gates from this wall. Additionally there are 12 Romanesque churches from the Middle Ages that worth looking at. Of course not all are original from this time, as Cologne was 95% destroyed during WWII. The Dom Forum also coordinates walking tours of these churches; although not sure now many are in English.
Cologne is a great city! It's not huge, so enjoy the smaller, intimate feel. And the people are generally friendly and helpful. Have fun!