Amplitude Modulation Circuit Components
Your biggest problem is C3. This capacitor has an impedance of about 47&Omega; @ 1000 Hz, which means that (ignoring V2 for the moment), the circuit has a gain of about -100 with respect to V1. With an input signal of 5V and a supply voltage of 15V, the output is going to be clipping pretty much all the time.

Try removing C3 altogether, and I think you'll find that the resulting output waveform is much closer to what you expect.

You may also need to tweak the bias resistor values. I think in this application, you need the transistor biased near cutoff, not at VCC/2 as you have it now.