The owner's manual tells you how to do it. You can get a pdf manual here
http://www.berettaservices.com/index.aspx?m=53&did=225
I'll give you a thumbnail outline of how to do it based on my familiarity with the 391 - as far as I know, the 390 works the same way.
Make sure the gun is cocked, and the safety on.
Remove the forearm nut, pull the forearm off, then pull the barrel off.
Remove the cocking handle from the bolt - just pull straight out.
Press the bolt release button while holding the sleeve around the magazine tube to keep the bolt from slamming shut (slamming shut is OK when the barrel and forearm are on, not when they are off).
Pull the sleeve around the magazine tube forward and off, and the bolt will come out with it. The bolt has 2 parts which will separate when out. Be careful not to lose the pin that holds the connecting rod on - it is loose enough to fall out.
Drive out the pin that holds the trigger group in the receiver. That is the pin that goes through the receiver just above the front of the trigger guard. If you don't have a proper punch, make one by filing the point off a nail, etc.
Hit the back of the trigger guard with the heel of your hand to start it moving forward a little bit. While holding the bolt release button down, pull the trigger guard (and the whole trigger group with it) forward and down, and it will all come out. Removing the trigger group is hard the first few times, but then it gets easier. The first time, it may help if you lay the gun on a padded table and get someone else to press the bolt release button and hold the gun in place while you pull the trigger group out.
Keep the hammer cocked while you spray clean the trigger group. That's why I said put the safety on before starting any of this. If you let the hammer go forward, a little pin in the side of the group will become loose and may fall out and get lost.
Someone more familiar with the 390 will have to tell you how to get access to the gas ports for cleaning - my 391 is different. Make sure you ream out the gas ports thoroughly - I use a 1/8" drill pit (held in my fingers, so I won't actually cut any metal) for my 391, but I don't know if the 390 has the same size ports.
I won't bother with additional instructions for removing the magazine spring or replacing the recoil spring - that needs to be done once in a while, but not real often.
I hope this helps.