You are asking about code points. In UTF-16 (C#'s char) there are only two possibilities:
- The character is from the Basic Multilingual Plane, and is encoded by a single code unit.
- The character is outside the BMP, and encoded using a surrogare high-low pair of code units
Therefore, assuming the string is valid, this returns an array of code points for a given string:
public static int[] ToCodePoints(string str)
{
    if (str == null)
        throw new ArgumentNullException("str");
    var codePoints = new List<int>(str.Length);
    for (int i = 0; i < str.Length; i++)
    {
        codePoints.Add(Char.ConvertToUtf32(str, i));
        if (Char.IsHighSurrogate(str[i]))
            i += 1;
    }
    return codePoints.ToArray();
}
An example with a surrogate pair ñ:
ToCodePoints("\U0001F300 El Ni\u006E\u0303o");                        //  El Niño
// { 0x1f300, 0x20, 0x45, 0x6c, 0x20, 0x4e, 0x69, 0x6e, 0x303, 0x6f } //    E l   N i n ̃◌ o
Here's another example. These two code points represents a 32th musical note with a staccato accent, both surrogate pairs:
ToCodePoints("\U0001D162\U0001D181");              // 
// { 0x1d162, 0x1d181 }                            //  ◌
When C-normalized, they are decomposed into a notehead, combining stem, combining flag and combining accent-staccato, all surrogate pairs:
ToCodePoints("\U0001D162\U0001D181".Normalize());  // 
// { 0x1d158, 0x1d165, 0x1d170, 0x1d181 }          //    ◌
Note that leppie's solution is not correct. The question is about code points, not text elements. A text element is a combination of code points that together form a single grapheme. For example, in the example above, the ñ in the string is represented by a Latin lowercase n followed by a combining tilde ̃◌. Leppie's solution discards any combining characters that cannot be normalized into a single code point.