FM modulator LTspice
The MV2201 has a nominal capacitance of 6.8 pF at 4 volt DC and can span a range of 1.9 to one when the DC control voltage changes from 1 volt to 10 volt. So, the range might be from 8 pF (@1 volt) to about 4 pF (@10 volts).
Now ask yourself how much that is going to change the main frequency of the oscillator given that the varactor diode is, in effect, in parallel with C22 and C23 (series elements). C22 and C23 are both circa 5 nF hence, the net capacitance to L1 is about 2.5 nF and you are expecting to &quot;observe&quot; a frequency deviation due to that capacitance (2.5 nF) rising a couple of pico farads and falling a couple of pico farads.
My calculations tell me that a 2 pF change in 2.5 nF will result in a frequency change of 1.0004 so if your centre frequency is 1000 Hz then I would expect to see it rise to 1000.4 Hz and fall to 999.6 Hz. Is your spectrum analyser able to spot this change? It's very small.
If you made L1 ten times bigger and C22 and C23 ten times smaller, the effect might be visible.

Firstly i don't get a clean sine wave on output.

Nobody expects it to have a clean frequency. The Colpitts oscillator is a phase shift oscillator and it doesn't run at the resonant frequency of the tank but to one side of it hence, the sinewave will always be distorted to some extent. The waveform I see in your picture is not unreasonable given your circuit.