Xan's on the right track for you here.
The Nonlinear platform will fit your equation, finding values for your parameters.
To start, you'll need to move all your parameters ("constants") as you described them, to one side of your equation. Essentially, you'll solve your equation for either C1 or C2.
I solved it for C1 here:
c1 = ((constant3 * c2)/(1-(1-constant3)*c2) * (constant2 - constant1)) + constant1
Then, in your existing data table create a column for a new "Predicted C1", using the Parameters option in the upper left of the Formula Editor to create your "constants".
Finally, use the Analyze -> Modeling -> Nonlinear menu to get to the Nonlinear platform.
You'll put your "Predicted C1" column in X, Predictor role, and your original C1 column in the Y, Response role.
Read more about the Nonlinear platform in the online documentation.
-Jeff