I have written a song...
It goes like this:
(4/4)  | Dm | F | C | Am |

Do you have any idea of how I can play the bass?
Like a melody or something...

I also play piano on this one, so it's kind of a ballad...


*The song begins with piano chords (as above).Then I play a melody (piano)
 Then the guitar fades in with the chords. And then, the chords change:
 | Dm | Em | Dm | Em | F | C | F | C | F | G |
 | Dm | F | C | G |...*

Could anyone find out a melody to play on the bass?

I shoud listen to the song to be able to give you an idea

What don't you simply root the chords with the bass and add a chromatic tone between chords?

The thing is...

I'm not much of a bass-player... 
I can learn it, but since I don't know what you're talking about...
I don't know how to "simply root the chords and add a chromatic tone between chords" 

Can anyone please explain that to me?

Like... I can read tab...

Heh. Dm, F, C, Am

You're playing the third, the seventh, and then the fifth (as I see it) of D.
 
D E F G A B C D E F G A B C

So, with your bass you can either play the root note (Of D is D, of F is F, of A is A, of C is C) and then the note between. D E F (On the bass, notes not chords) But when going to A from C, I am saying play C A# A.
Pretty simple but it works for your purposes.

[edit] And try to swing the notes in the transition, hold the first one longer then play the last two rapidly.)