If time stops inside a black hole, due to gravitational time dilation, how can it's life end after a very long time? If time doesn't pass inside a black hole, then an event to occur inside a black hole needs infinite time relative to the outside. Thus, it will never age.
Please keep it simple..
I assume you're asking how a black hole can evaporate due to Hawking radiation. The answer is that the Hawking radiation does not come from the event horizon, but instead comes from a region just outside the event horizon so time has not stopped at its position.
If you were to watch a black hole form then evaporate, you would never see an event horizon form. That's because in your co-ordinates the event horizon would take an infinite time to form. You would see the infalling matter slow and red shift, then be re-emitted as Hawking radiation without the event horizon ever having formed.
The time dilation you speak of is a description of the apparent time an observer outside witnesses someone falling into a black hole. That is, if you are standing outside the black hole (some distance away) and you watch you buddy jump in, it will look at if he slows down as he reaches the horizon, and never quite gets there. As for your buddy, he sees his watch ticking away normally and he notices nothing special as he crosses the horizon.
This is just another specific case of one observer noticing how the clocks in another observers reference frame appear to tick. The same is the case for 2 people moving in flat space at constant velocity with respect to one another. Each observer sees the others clocks running slower, but sees their own clocks run 'normally'.
Please check out the final paragraph in section 1.1 "General Relativity" en.wikipedia.org/wiki/Black_hole If time inside a singularity is not moving to the outside observer which is what matters. And which is the frame at which we are measuring the 10^100 anticipated years for the BH life to end, but then we have to wait forever for any aging to occur.. – Force 9 hours ago
Time does not stop inside a black hole. As others pointed out already, for the observer inside a black hole, time simply goes on.
Time does not stop for observers outside of the black hole "looking in", either. It merely slows down to an extreeeeeemly slow crawl. That's a subtle but important difference. Time keeps ticking away, but it is infinitely (no pun intended) slow: It is so slow that it would take an infinite amount of time before the outside observer can see someone actually fall in. The fact that it takes an infinite time however does not mean that time stops.
Say you're a distant observer having your own clock and you're observing a clock falling into the black hole. As the clock approaches the black hole, the time measured by you is much slower than your own clock. As it nears the event-horizon, it gets red-shifted so much and at its closest distance to the horizon (somehow you managed to see the red-shifted clock), you'll see the clock freeze and stands still there. Because, further photons can't escape from the black hole's strong curvature and you'll see the photons that are trying to escape from the blackhole. This results in a scene that you'll see the clock stay there forever, never reaching the event-horizon. (The same thing happens during the formation of a blackhole)
But if you are an "in-falling observer" along with the clock, you'll see that your clock is normal. Even after you've fallen into the blackhole, you can see your own clock tick. But, you can't quite determine whether you've crossed the horizon or not..! So, this kinda (local) observation does not account for the blackhole's age anyway.
On the other hand - if you're interested in the age of blackhole, Hawking radiation is useful to determine (as mentioned by John). In its principle, the blackhole evaporates by quantum gravitational effects which makes use of virtual particle pairs and the rate of this emission is proportional to the mass. But, the problem here is (due to quantum scale) that the evaporation is very very low (not simply some billion billion years as for other phases of stars)...
