I started reading Kitzur Shulchan Aruch. In the second chapter I noticed that I have to clean my hands when I wake up, then I say a blessing (netilat yadayim). Am I allowed to use soap with water when I wash with my hands, or should it be pure water in a washing cup?
This Q & A summarizes the rule and its footnotes lead you to the discussion in Shulchan Aruch with more info.
Citing from it:
One must be careful that one’s hands are free of dirt that one is generally makpid (particular) about. Even if one is not particular about the type of dirt, but people in general are particular about the type of dirt, one must also ensure that hands are clean of it before washing. It therefore depends on the particles in question.
For ink marks, although most people are probably particular, stains that have no substance do not constitute a chatzitzah (Mishnah Berurah 161:14).
If one washed without cleaning off dirt that one is not particular about, but that most people are particular about, one should wash again without a berachah, because this question depends on a dispute among poskim.
See dispute between Rambam (personal, subjective), and Rashba (objective) concerning the question of dirt that most are particular about and the person in question is not particular. We are concerned for both opinions on a lechatchilah level (Mishnah Berurah 161:7, 10, citing from the Magen Avraham).
Based on the bold phrase it seems that the rule is if you wake up and for some reason, you have dirt on your hands, (it DOES happen for various reasons, that I don't care to elaborate, here.) then, you need to wash your hands with soap first.
However, as I have stated in my comment, netillat yadayim is a separate process which means that you MUST use the cup. (See Shulchan Aruch which discusses the process and procedure.) The point is, that washing your hands with soap if there is dirt is a prerequisite to netilat yadayim, not a substitution.