What are the weather requirements to log an instrument approach?
Update with relevant info from InFO 15012 (I'm skipping the FTD portions as not relevant to the question, and including the simulated parts but italicizing them as they're useful but also not relevant). Translated, you can log an approach when:


it is flown solely by reference to instruments, and


it is flown in IMC, and Actual Instrument time is logged, or
the PF is using a view-limiting device, and Simulated Instrument time is logged;

you must fly the complete approach from a feeder or the IAP (or Vectors to Final) and remain established the whole time. This means you fly initial, intermediate, and final segments, unless you're on vectors.


if flying the approach in IMC, you may log an approach that:


continues to DA/DH or MDA in actual, or 
becomes VMC before DA/DH or MDA, provided you are inside the FAF.

if simulating the approach, you must continue to the DA/DH or MDA under the hood (or other VLD);



That's pretty much it. Here's the source text:


  A pilot may log an IAP for currency or training when the pilot
  accomplishes the IAP in accordance with the following conditions:
  
  
  When conducted in an aircraft, flight simulator, flight training device, or aviation training device, the pilot must operate that
  aircraft or authorized training device solely by reference to
  instruments [§ 61.51(g)(1)];
  When conducted in an aircraft, flight simulator, flight training device, or aviation training device, the pilot must be established on
  each required segment of the IAP to the minimum descent altitude (MDA)
  or decision altitude/decision height (DA/DH);
  When conducted in an aircraft simulating instrument flight conditions, a flight simulator, a flight training device, or an
  aviation training device, the simulated instrument meteorological
  conditions (IMC) must continue to MDA or DA/DH; and
  When conducted in an aircraft, the flight must be conducted under actual or simulated instrument flight conditions [§ 61.51(g)(1)].
  When conducted in an aircraft maneuvering in IMC, and the aircraft transitions from IMC to visual flight conditions on the final approach
  segment of the IAP prior to or upon reaching MDA or DA/DH.
  




Previously, I linked to Rod Machado, who cites the FAA in an answer to this that I quite like, and which was pretty frequently referred to when I was instructing at a flight school:


  Once you have been cleared for and have initiated an instrument approach in IMC, you may log that approach for instrument currency, regardless of the altitude at which you break out of the clouds.


...but this is not the latest knowledge anymore. It's functionally pretty close to the InFo rules for IMC approaches, though!