To remove the dust, you will need to remove the lens and disassemble it in a cleanroom. You’ll need to clean both, and inspect all of the elements to do it correctly. CAUTION: Unless you can line up the sensor correctly, the image will never be 100% correct again. You are taking a risk by removing it, as it can/will permanently ruin the factory alignment!
Using a cleanroom for this is essential. If you do it in a normal space, the problem will be worse.
Generally speaking, if the lens dust is visible, it tends to be on or very close to the sensor. If the dust is somewhere else, it isn't as likely to affect the image, unless it’s particularly bad. To remove the lens assembly from the camera, here is the guide: Panasonic Lumix DMC-FX01 Lens Assembly Replacement. For additional reference, here is the service manual.