Instructions
Go to Goldenrod City and go into the Goldenrod Underground area. There is only one person there. They will give you a Rocket disguise you can use to infiltrate the Radio Tower.
Go to the Radio Tower and talk to the Grunt at the door. You will have to battle him when your Rival blows your cover -- he has two Raticates, Normal-type Pokemon which you can easily beat with Fighting-type attacks. After beating him go into the Tower and head up to the fifth floor.
Go into the Radio Director's Office and you will meet Rocket Executive Petrel. He has six Pokemon, but five of them are Level 30 Koffings. This is a Poison-type Pokemon which is weak against Ground- and Psychic-type attacks. His other Pokemon is a Level 32 Weezing, an evolved version of Koffing with the same weaknesses. After you beat Petrel he will give you a key.
Leave the Radio Tower and go back to the Goldenrod Underground. Your new key opens the door at the southern end of the Underground and, after using it, your Rival will show up demanding a battle. He has a Level 32 Golbat, which is weak to Electric- and Grass-type attacks; a Level 30 Magnemite, which is weak against Ground-, Fire- and Fighting-types; a Level 32 Haunter, weak to Ghost-, Psychic- and Dark-types; a Level 34 Sneasel, which is weak against Fighting-, Fire-, Bug- and Steel-types; and a Level 34 Ouilava, which you can beat with Water- or Rock-type attacks.
Move through the locked door and you will find a puzzle involving three buttons. The correct sequence is blue, green, red -- just work from left to right. At the end of this sequence, you'll be able to free the Radio Director and you'll emerge into the basement of the Department Store. Get the Card Key from the Director, leave the store and head back to the Radio Tower.
Head to the third floor of the Tower and go through the previously locked door. Here you'll meet Rocket Executive Proton. He only has two Pokemon, a Level 33 Weezing and a Level 28 Golbat.
Beat Proton and move on up to the fifth floor. Here, you'll battle Rocket Executive Ariana. She has a Level 32 Arbok, a Poison-type Pokemon which is weak against Ground- and Psychic-types; a Level 32 Vileplume, a Grass/Poison-type that you can beat using Fire-, Flying-, Ice- and Psychic-type attacks; and a Level 32 Murkrow, a Dark/Flying-type which is weak against Electric-, Ice- and Ground-types.
Go up to the top floor in the elevator Ariana was guarding, and you will meet the final Rocket boss, Executive Archer. He has a Level 35 Houndour, a Dark/Fire type which is weak against Fighting-, Ground-, Rock- and Water-types; a Level 35 Koffing; and a Level 38 Houndoom, an evolved form of Houndour which has the same weaknesses. Beat Archer to complete the liberation of the Radio Tower.