My best grades by University
by? in university would probably be better, unless I got you wrong
All of them I liked very much and tried to find extra information in addition to primary literature.
this sounds kind of shallow and banal. maybe say more about how you tried to gain a deeper understanding of the subjects and explored additional literature?
In the period of study, without interrupting,
I would change that to something like "during my studies, in addition to attending classes"
diploma on the additional qualification of an English interpreter in the direction of biology.
this sounds a bit awkward. you might want to find the official name of your program to make it sound more correct.
I think philological education gave me not only knowledge in English but also contribute to my personal development.
think is a way overused word. "believe" or "feel" might be better. also, "knowledge of English" and "contributed".
the part where you describe the work you did sounds great and very impressive, at least to me
On my opinion control of gene expression is one of the most (very) existing and perspective theme and I want to develop my knowledge in this area that is why this position interested me. Although I want to contribute to work on gene regulatory networks underlying hematopoiesis and regulatory disturbance leading to leukemia.
I am enclosing my Curriculum Vitae and details of my academic referees. If you require any additional materials of information, I am happy to supply it.
"very existing"? i'd say "highly researched" or "important". and you might want to call it something different than a "theme" - a topic, an area of biology, an area of genetics, a problem in genetics etc.
"I will be happy to provide them"
overall, good work! you might want to polish it a few more times, maybe give it to your English tutor or your English-speaking friends to check the grammar once again, because you can't really say you have a diploma of an interpreter and have poor language skills.