How do I drive 14.3Mhz clock input from 10MHz?

What you need is a PLL, a phase-locked loop. It works by comparing one oscillator that you can control, with a reference oscillator. The trick is that it is easy to divide the frequency of an oscillator using a digital counter, so what you do here is to divide the 14.3 MHz oscillator by 143, the 10.0 MHz reference by 100, and then use the output from this comparison to make sure that the 14.3 source is running at an exact relation to the stable 10 MHz reference.

There are numerous circuits that can do all of this in one package, sometimes even including a reference oscillator. It is very common having to synthesize frequencies from a stable oscillator, so these are not unusual.

It is possible to change the order of multiplications and divisons to avoid frequencies above \$100~\text{MHz}\$. If you want a pretty square wave, the last step should be a divison by \$2\$.

\begin{align*}\frac{10~\text{MHz}}{2} &= 5~\text{MHz} \\ 5~\text{MHz} \cdot 9 &= 45~\text{MHz} \\ \frac{45~\text{MHz}}{11} &= 4.090909~\text{MHz} \\ 4.090909~\text{MHz} \cdot 7 &= 28.636363~\text{MHz} \\ \frac{28.636363~\text{MHz}}{2} &= 14.3181818~\text{MHz}\end{align*}

If you want 14.31818181818 MHz from a source of 10 MHz, it is difficult. The 14.31818 MHz is the american TV color burst frequency, the precise value is 315/22 MHz. You may divide 10 MHz by 2, multiply by 9 and by 7 to get 315 MHz. Then you divide by 22 to get the frequency you want. May be more than one PLL is necessary to do that. Another way is to divide the 10 MHz by 4 and multiply by 9 and 7 and finally divide by 11.

Of course it is theoretically possible to multiply by 63 and then divide by 44. But this requires a very fast PLL oscillator for 630 MHz and also a fast frequency divider. I suggest to divide by 22 first, then multiply with 63 and finally divide by 2. But for a low phase jitter, separate multiplications by 9 and 7 may be better.