Distortion Festival is a 5 days big party, rolling through one neighbourhood each day during the first week of june. In the daytime you'll find free street parties and at night the parties continue in the night clubs with international DJ line-ups.
In the summertime you can take the Metro directly to the beach. And if you're over the crowded white trash tanning and blanket sex scene, enter the hidden away mens or womens area located in the big wooden beach bathhouse. Here you can lie naked and enjoy the sun on your private parts while reading a novel or surfing your iPad. Expect lots of gays and lesbos which can be a curse or a party depending on who you are as a person.
As the summer is my favorite time of the year and that the beach is just 15 min bikeride away, I recommend Amager strand. Full of people usually though, but even so it gives a bit of an exotic touch to such a scandinavian city with the sea and the bright sand in the hot summer days.
Copenhagens bikelanes is all around the city. Its one of the best cities in the world to get around on bikes, there is even free bike servicestations around town:)