I recently bought Diana and did well with her, but some times my q never registers and I'm left under a tower of a group of champions without a second ult. Is this because of my personal skill or the server lag?

I play her often and haven't noticed, so unless it's very new I bet it's your internet connection. Check your upload speed, 95% of your lag will come from a low upload speed, your download will be more than sufficient even with a very low download speed.

Thanks man!