The answer to this depends more on how university years are organised in your region. University years are not uniformly structured across the English-speaking world.
A fairly generic approach might be:
How far along are you in your university programme?
What year (of university) are you in?
In much of the Commonwealth (including Canada), the answer will likely include year of the programme a student is in. There might be a reference to the school semester or term.
- Winter (usually January-April)
- Summer (usually May-August and sometimes a part-time/wrap-up period at many schools)
- Fall (usually September-December)
I am just finishing up my second year this Summer.
I will start my final year this Fall.
Americans often use "freshman", "sophomore", "junior" or "senior" to refer to first, second, third or fourth year respectively. This is true for both high school and college/university in the US. These terms are not commonly used outside the US.
Wikipedia has a long article on university terms across the world, but some of the information may be generalised.