How to deal with graduate school rejection and overall academic failure?
I also did my undergrad in the most prestigious university in my country (a developing country) albeit in engineering and started graduate school in the U.S. I have some suggestions for you.

First of all, you need to apply for about 10-20 schools to have a chance. Do not focus on the most prestigious universities (Harvard, MIT, etc.). This was something I had issues with, I didn't want to settle for less than the best because I was in the best university in my country. The truth is, at least in the U.S., there are many many good universities.

Do your research and find schools which you would think would be a good match for you. To get a start you can look at a ranking for your major. If you are looking at the U.S. there will be graduate school rankings for your major published by U.S. news. They can give you a good idea of which schools you may have a better chance of getting into. Once you find the programs you are interested in look up professors and research groups in those universities and even try to get in touch with them and let them know you are interested in their work. Don't just blindly email them, familiarize yourself with their research, read their publications, and try to discuss their work in your emails. Be specific about what you like and what you are interested in and they may help you out and give you suggestions to steer you in the right direction. Take note that professors receive tons of emails from prospective students on a daily basis and often won't read them, so don't get discouraged if they don't respond.

Finally, something I did not see you mention was standardized exams. If your undergrad classes were not taught in English you would have to take the TOEFL. Most U.S. universities also require taking the GRE general exam. For biology, you may also have to take the subject test as well. Focus on getting good scores in all of these tests. Even in the sections not related to your major such as the verbal section of the GRE general test. Often grad schools will be reluctant to offer admission to students from foreign countries who do not have good verbal scores even if for majors like biology.

If this is really your dream, don't give up. Good luck :)