AM (Amplitude Modulation) Colpitts
You can't push a colpitts oscillator (or hartley or clapp) too far because you will either get an overly distorted sine wave or you'll kill oscillations completely. Even if distortion wasn't too bad (i.e. you didn't push things too much) there would be an associated frequency modulation due to the changing bias conditions brought about by amplitude changes. 

The underlying mechanism here is the so-called "miller" capacitance between base and collector - basically the depletion layer in the PN junction in that part of the BJT is modulated by the voltage across collector and base. In fact any oscillator of this type produces a cyclic distortion that is related to the change in capacitance due to the actual oscillation voltage appearing between base and collector.

So, my advice is add an amplitude modulator to the output of the colpitts and this can be easily done with a diode (and the appropriate DC control levels) plus an output filter resonant at the carrier frequency. Here's a very simple AM circuit idea you can experiment with: -



The blue waveform is the modulated carrier and the red signal is a triangle wave modulation signal. You can get quite respectable results with virtually a really small handful of components.