This answer should probably be shortened and simplified a bit.
Won't extended life worsen overpopulation problems?
Overpopulation is something we would have to come to terms with even if life-extension were not to happen. Some people blame technology for having given rise to the problem of overpopulation. Another way of looking at it is to consider that were it not for technology then most people alive today would not have existed -- including the ones that are complaining about overpopulation! Were we to stop using modern agrecultural technologies, such as tractors, fertilizers and pesticides, most humans would soon die of starvation. It's worth thinking twice before calling something a "problem" when owe our very existence to it.
This is in no way to deny that too rapid population growth causes crowding, poverty and depletion of natural resources. In this sense there is a very real problem. Efforts to promote education about family-planning and contraception, especially in the poorest countries (where population growth is fastest), should be vigorously supported. The constant attempts by some religious pressure groups in the United States to block these humanitarian efforts are seriously misguided in the opinion of transhumanists.
We can also be hopeful that scientists will be able to keep up with the increasing demands of a growing world population. For example, we have just begun unlocking the potential of genetic engineering. (Cloning is just one very helpful method in genetic engineering.) It's impossible to foretell exactly how far that will take us, but it is already clear that it will enable us, among other things, to substantially increase crop yield and effectiveness in animal husbandry. If we could design cattle without brains or with just the brain stem, we would have a way of producing meat without maltreating animals (the brainless bodies wouldn't count as animals). The human yuck-feeling (probably temporary) would have to be weighed against the permanent reduction in animal suffering.
One thing that the environmentalists are right about is that the status quo is unsustainable. Things cannot, as a matter of physical necessity, remain the way they are today indefinitely or even for very long. If we continue to use up resources like we currently do then we will run into serious shortages sometime in the first half of the next century that will force the world economy to contract and the world population to decrease drastically. The deep greens have an answer to this: they suggest we try to turn the clock back and return to an idyllic pre-industrial age in harmony with nature. The problem is that the pre-industrial age was anything but idyllic - poverty, misery, disease, heavy manual toil from dawn to dusk, superstitious fear and cultural parochialism. Also, it's hard to see how more than a few hundred million people could be maintained at a reasonable standard of living with pre-industrial production methods, so 90% of the world population would somehow have to get rid of.
Transhumanists propose a much more realistic alternative: not to go backward but to push ahead as hard as we can. The environmental problems that technology creates are problems of intermediary, inefficient technology. Technologically less advanced industries in the former Soviet-block pollute much more than their Western counterparts. High-tech industry is relatively benign. When we develop molecular nanotechnology we will not only have perfectly clean and efficient production of most any commodity but we will also be able to clean up the mess created by today's crude production methods. This sets a standard for a clean environment that transhumanists challenge any environmentalist to try to match.
Nanotechnolgy will also make it cheap to colonize space. From a cosmic point-of-view, Earth is a totally insignificant little speck. It has been suggested that we ought to leave space untouched and preserve it in its pristine glory. This view is hard to take seriously. Every hour, through entirely natural processes, huge amounts of resources - thousands of times more than the total of what the human species has used throughout its career - are transformed into radioactive substances or wasted as radiation escaping into intergalactic space. Whoever was in charge of this project could probably have done with a little bit more accountability!
Even with full-blown space colonization, however, population growth can continue to be a problem. If the expansion speed is limited by the speed of light then the amount of resources under human control will only grow polynomially. Population on the other hand can easily grow exponentially. If that happens then the average income will eventually be drop to the Malthusian subsistence level, which will force population growth to slow down. How soon this would happen depends primarily on reproduction rates. Increases in average life span does not have a big effect. Even vastly improved technology can only postpone the inevitable for a relatively brief time. The only long-term solution is population control restricting the number new persons created per year. This does not mean that population could not grow; only that the growth would have to be polynomial rather than exponential.