Vilnius is the capital of a country that once reached from the Baltic to the Black Sea, and it has the Old Town to show for it: one of the largest in Northern Europe, UNESCO-listed since 1994, and so full of baroque churches that the style is called Vilnian Baroque. The city has also been ruled by Poland, Russia, Germany and the Soviet Union, and before the Second World War it was one of Europe's most important Jewish centers. Most of that Jewish city was murdered between 1941 and 1944. You'll see both the beauty and the loss here, and the guide doesn't skip either.
Vilnius is compact and easy. The Old Town can be crossed on foot in half an hour, and the people are friendly in a reserved Baltic way. Five days is the right length because the interesting parts sit in different corners: the castle hill, the little republic across the river, the former KGB headquarters, the places of Jewish Vilnius and a castle on a lake outside town. Add the food, which is heavy, potato-based and better than its reputation, and you have a plan that fills five days without rushing.
The plan runs day by day. Day one is Cathedral Square, the Palace of the Grand Dukes, the university and Gediminas Tower. Day two is the Gate of Dawn, Halės market and Užupis. Day three is Jewish Vilnius and the Museum of Occupations, day four is Trakai Island Castle, and day five covers Paneriai, the 1991 barricades and the TV Tower. Kaunas and the Hill of Crosses are in the "more time" section. Start on a Wednesday: the Palace of the Grand Dukes and the Jewish museums are closed on Mondays, the Museum of Occupations is closed on Mondays and Tuesdays, the university courtyards and the cathedral bell tower are closed on Sundays, and Trakai's castle is closed on Mondays, all of which this order respects. For the country around it, see Lithuania. If you're continuing north, 5 days in Riga and 5 days in Tallinn follow the same pattern.