If no soul may see God unless it has been purified which is why we must go to Purgatory before Heaven; and if Purgatory ends on Judgment Day, what happens to the souls of the people who are still alive on Judgment Day? How are their souls purified?

Actually, we don't know if purgatory ends on Judgment Day. It could, arguably, continue for some "time" thereafter. Then again, maybe not, perhaps the Last Judgment ushers in a quick searing purification and purgatory passes away with the current heavens and earth. 
 
But perhaps a more fundamental "answer" to your question is to say that there are just some things we don't know. Good Theology must recognize its limits, being content to accept that there are many things God has not revealed. And even in those things He has revealed, we must humbly admit that the mysteries about God and creation have depths beyond our capacity to fully comprehend.