To know your Moon and the rest of your planetary influences, you need to get your birth chart done: here.
You Sun is in Gemini, and if you were born June 4th, you were born on the 2nd decan of Gemini. Geminis from this “week” or “season” (that goes from June 1st to June 11) are not only ruled by Mercury, the planet of communicacion, but also by Venus, the planet of love; and some astrologers believe that also by Mars, the planet of war and passion.
The rest of your birth chart (rising sign, moon, etc) can completely contradict the influences from this planets but still, this is the general knowledge about your decan:
The influence of Venus makes your relationships with others extremely important. 2nd decan Geminis hate to be alone and are something of “closet romantics”. These natives yearn for lasting love and have a good chance of finding it, assuming that the inherent fears associated with this Decan can be put aside (Libra fears to be alone). These natives are effective speakers who can communicate information in both a clear and convincing manner. In general, they insist upon harmony and good taste (specially about fashion). The disposition tends toward being positive and friendly, but if the influence of Mars (the other secondary planet governing this Decan) is overly-prominent, then these natives may be more aggressive.
In this instance, the versatile Gemini nature will display a powerful and unexpected drive (like it does in Aries or Scorpio, the other planets ruled by Mars). Although this variation of the Second Decan is undoubtedly Gemini in character, it will be one that is “stepped-up” to such a high degree that it could reach the point of ruthlessness. Here, the quick and scheming ways of Mercury are “hitched” to the bull-dozing tactics of Mars, but may well be directed toward good instead of wasteful purposes…particularly if the sympathetic understanding of Venus is cultivated (activities that strengthen your relationship with beauty, nature and art).
If im not mistaken, you share your birthday with Angelina Jolie. Besides that, you have the same traits that every other Gemini has.