When do pilots perform a CAT II/III autoland?
CAT II / III operations depend on DH and RVR, not cloud layer. It will be based on "the most demanding" of the two.*

Extreme low visibility, say due to dense fog (which is a cloud type), can actually be considered OVC000, hence the use of RVR. RVR tells you how bad the fog is and how far you can see inside it.

Here's an example of such METAR for Charles de Gaulle Airport:


  LFPG 051100Z 13005KT 090V180 0350 R27L/0900U R09R/1500D R26R/0800N R08L/0750N R26L/0800U R08R/0650N R27R/1000U R09L/0800U FG OVC000 00/00 Q1033 NOSIG


The airport also needs to enforce low visibility procedures (LVP) and broadcast it on the ATIS, so the ILS' safety area is protected. By planning ahead and by checking the ATIS, the pilots will know which approach to expect based on their operator's established 'operating minima'.

There's more to it however than DH, RVR, and LVP, for further reading check the Airbus publication, 'Getting to grips with CAT II / CAT III operations'.



* ICAO Manual of All-Weather Operations

Re: comment:


  
    
    Can cat II/III only be done, when airport announced (ex. via ATIS) that LVP are in force?
    
  


Yes, the alternative is for the pilot to request from the tower/approach a CAT II / III approach, so the tower can enforce the safety area and verify the ILS equipment is working up to the CAT II / III standards.


  
    
    If the weather was like in your METAR - RVR0800, the pilot would have to use CAT I minima if no LVP is enforced, but he could use use Cat II/III only if LVP is in force to make sure he won't have to go around - is that correct?
    
  


Without checking the French authority's specifics, the METAR visibility (350 m) is already below CAT I (>800 m), so the airport should be operating in LVP.


  
    
    (...) after choosing CAT II/III he must auto land - is that correct?
    
  


It will be based on their operator's established 'operating minima', i.e., the airline procedures, which adhere to the regulations set by its national authority. CAT III B must be flown using auto land to touchdown incl. roll-out. CAT II and III A not necessarily.