Here’s a probably stupid question:
Suppose you’re writing something in C major, which is all natural notes. Then you choose to take a walk on the wild side and play a forbidden black key. Do you write D# or Eb?
It depends.
If the note is part of a chord, you use the appropriate notation - for example, if it’s F7, Eb is the correct notation. If it’s not part of a harmony, but an ornamentation to the melody, you use whichever is convenient…a rising pattern would be D-D#-E, a descding one E-Eb-D, each of which avoids having to have a natural to cancel the preceding accidental.
Exactly. Ease of reading is extremely important.
Although the O.P. is talking about tonal music in C Major, you should also consider proper note spelling in atonal music.
Generally, if an interval is supposed to sound like a third, it should also look like a third. If you write C-D# (which by itself sounds like a minor third), you better have a good reason for not writing Eb. There are many of these reasons, but they all basically boil down to context.
Seems to me the OP is talking about the number of sharps and flats when writing musical keys. In which case, the convention is:
Sharps:
[ol]
[li]G major[/li][li]D major[/li][li]A major[/li][li]E major[/li][li]B major[/li][li]F# major[/li][li]C# major[/li][/ol]
Flats:
[ol]
[li]F major[/li][li]Bb major[/li][li]Eb major[/li][li]Ab major[/li][li]Db major[/li][li]Gb major[/li][li]Cb major[/li][/ol]
As there isn’t a key of D# major, you put down three b’s for the key of Eb major. For some equivalent keys like Cb major (seven sharps) and B major (five flats), you go with the simpler-looking one: B major.
To expand somewhat (and I understand the question to be how to notate accidentals in the key of C, not a key signature question):
This is where a little theory comes in handy. (And this is basically what GorillaMan has already said. If the note is a melodic passing tone, then it is notated whatever is simplest (usually a sharp when going up and a flat when going down.) If the note, say the D#/Eb in the key of C is playing a harmonic function, then it depends what that function is. In the context of a G+ (augmented) chord in the key of C, it would be notated as a D#, since in an augmented chord, the fifth is raised a half stop. If the Eb were part of a Gflat diminished chord (which probably won’t happen in C), it might even be notated as an F double flat.
The rules aren’t always clear-cut. For example, one song that comes to mind is the notation of the bass line of the Beatles’ “Lady Madonna.” The first two measures of the bass line are: A A B#/C C# | D D F# A. It’s that third note that causes orthographic problems. Do you hear it as an ornamental note, a passing tone, or appogiatura, that pushes into the C#? If so, then the B# notation would be your perference. Or do you hear it as a minor-major shift–which is not uncommon in blues and rock–and therefore C natural would be the appropriate notation?
Actually, Terminus, you completely missed the point. 
The question was about which accidentals to use when there is no key signature. (Or an “empty” key signature, as it were, which is the case in C major.)
Ah, in that case, you go with what everyone else has already said.
Nitpick: D# major does exist. But it’s so far round the circle that it would have double-sharps in the key signature, which ain’t too friendly on the eye.