I don't think we're going to find any evidence (like TimB did for this question) that life does exist on extrasolar gas giants, but we can at least use some logic to figure it out.
First, the cores of gas giants are nothing like asteroids, comets, or even terrestrial planets. Jupiter's core is thought to consist of rock, surrounded by metallic hydrogen - in a liquid state, known as a supercritical fluid. Temperatures there are thought to reach a whopping 36,000 K - just at the outer edges - and pressures are thought to be about 3,000,000 Pascals, much greater than on Earth. Here's a guide to the interior of Jupiter (way too warped to put here).
Saturn's core isn't much different: a ball of rock and ice surrounded by hydrogen and helium at enormous temperatures and pressures. Uranus's core is only at about 5,000 K and a measely 800,000 Pascals, but that's still very harsh conditions - too harsh for life to survive. Neptune is much the same. Any extrasolar gas giants probably have very similar internal structures.
there could be a layer of atmosphere similar to that of earths.
No. The conditions anywhere near the core are fairly bad.
Also what would it take to leave the atmosphere?
You're looking for escape velocity. At a distance $r$ from a body with mass $M$, the escape velocity is
Given that the cores are generally many more times massive than Earth (though Neptune and Uranus may be exceptions) and are also bigger than Earth, I'd say that the answer would be that it would be very, very hard (i.e. impossible), taking into consideration the temperatures and pressures down there.
Assuming, however, that there somehow is life in the core of this planet, and some of the life forms want to get off the core, here's what they'd face when trying t build and fly a spaceship off the planet:
- Temperature. Iron melts at 1811 K and boils at 3134 K - and the temperatures here are an order of magnitude higher than that! I haven't been able to find figures for steel, but I'm guessing it wouldn't do you any good here. Copper, brass, and just about any other metal you can think of would most likely be in some sort of gaseous state.
- Pressure. Same problems here as with pressure. Landers on Venus, with its surface air pressure of 9.2 MPa, have been crushed in a very short period of time. If the temperature didn't get any materials on Jupiter's core, the pressure would.
- Drag. Materials aside, you're going to have to get through tens of thousands of miles of gases here. I can't find the exact figures for the atmospheric gases, but I would assume that their density would make it very tough to get anywhere. A lot of energy would be lost to atmospheric friction, not only taking away energy but heating up the spacecraft.
TimB recently brought up a good point in a comment:
Hmm, the thing this answer misses is that there is a band as you come further out of the atmosphere where the pressure and temperature are actually reasonable. The question then arises as to whether life could evolve there.
We did have a question a while back about whether or not life could form on a planet with multiple gas layers. There was a division over this: TimB said it was unlikely for anything bigger than bacteria, and I came up with a whimsical scenario involving pufferfish polyps.
Bacteria could definitely live up somewhere in the upper atmosphere of a gas giant. I don't think there's any doubt about that. But any life in the gas giant's atmosphere would have to be high up, and therefore either flying or lighter than air. The first is possible but only if the creature was introduced pre-evolved. The second is less likely. Bacteria are your best shot.
Also, the life-forms would have to breathe an odd mixture of gases. Their best bet would be hydrogen and helium, neither of which is conducive to life, although they could also go for methane, trace amounts of oxygen, or ammonia-based compounds. There's also a bit of water, although that would be useless for respiration. Quite frankly, I don't think complex life could evolve in the atmosphere - or anywhere on a gas giant!