Converting IQ to RF data
For ideal performance, you can DFT your input, pad out the spectrum with zeros, and inverse DFT the result.  This is not a particularly efficient approach, though.

Pretty much the same result is obtained with interpolation by zero-padding and then running that stream through a low pass filter.  An FIR sinc filter is preferred for this for two reasons--(1) it has nearly ideal performance and introduces negligible distortion into the baseband signal, and (2) when zero padding, most of your input samples are zero.  Since 0*x=0, you don't have to do all those multiplies; for your 6:1 example, you can design a 72-point FIR and still only do 12 multiplies per output sample.

Once the filtering is done, you can upconvert (multiply by the exponential).  This should be $e^{j\omega t}$ where $\omega=2\pi f$ and f is your RF carrier frequency.