What is the best hash function for uint64_t keys ranging from 0 to its max value?

If all you have to hash is a uint64_t of any possible value with unknown probabilities, and your output must be a uint64_t, then you don't gain any advantage by changing the value. Just use the key itself.

If you knew something about the distribution of your values or your values were restricted to a smaller range (which is really the same thing as knowing about the distribution), then it could be beneficial to apply a transformation to the key, but this depends on the implementation of the container. You would only benefit by reducing collisions when the table transforms a hash into a bucket index, but that depends both on the table's algorithm and the current/average state of the table (how often each bucket is used).

I would suggest a good 64-bit mixer of which there are many to choose from. The finalizer from MurmerHash3 is fairly quick and does a reasonable job in just five lines of code:

key ^= key >> 33;
key *= 0xff51afd7ed558ccd;
key ^= key >> 33;
key *= 0xc4ceb9fe1a85ec53;
key ^= key >> 33;

Numerical Recipes, 3rd Edition, recommends this:

public static UInt64 Next( UInt64 u )
  {
  UInt64 v = u * 3935559000370003845 + 2691343689449507681;

  v ^= v >> 21;
  v ^= v << 37;
  v ^= v >>  4;

  v *= 4768777513237032717;

  v ^= v << 20;
  v ^= v >> 41;
  v ^= v <<  5;

  return v;
  }