I wanted to apply PhD in Clinical Psychology, but my research activities and research interests are a bit different A bit background about myself after finishing my bachelor degree in Psychology from a private university in a developing country. I worked at my family business for a couple of years before realizing that I wanted to become a clinical psychology scientist and part-time practitioner.

Research activities in clinical psychology field in my country is sadly not too active and rigorous compared to other branch of psychology. Most of the lecturers in clincial psychology are much interested to become a practitioner, thus research activities was so low in the clinical psychology field. However, luckily, by the time I became interested in the clinical psychology research, a suicide prevention (SP) organization with quite high research activity was founded and I became a part-time researcher for the organization.

Currently, I am planning to apply for a PhD program in Clinical Psychology in the West (specifically USA or Canada, because UK, Australia, and New Zealand PhD Clinical Psych programs rejects bachelor holders from non-English speaking countries, I cannot speak German or other European native language, so I will not apply there) because the reason mentioned above. However, my biggest problem is that I wanted to do research in marital and sexuality issues, but most of my research experience is heavily related to suicide topics. What should I do?

The good news is recently a lecturer from a different from mine graduated from a PhD Clinical Psychology abroad and she seems to have certain interest in marital issues topics. Should I approach her to gain up experience in conducting research in that topics? However, I am rather difficult position because I have to balance my obligation in my family business and SP organization. Also, she seems to be very busy and she said that it would be rather difficult to accomodate my research project.
I don't know anything about your own educational system and I can't speak for "the West" in general, but in the US you shouldn't have any particular difficulty, other things being equal. In fact, having some research experience in the general psychology area would likely be treated as a plus no matter what the focus. 

In the US, it is possible, in general to be accepted into doctoral programs with only an undergraduate education and very little if any research experience. You also don't, in general, need to specify a specialty on entry, but choose one during the first years. Such programs normally have some course work to fill in gaps. 

The path, for the US, actually, is to apply to programs that interest  you, or, at least, contact departments at such institutions and discuss your qualifications and interests. 

The European system is different, and I'll let other speak about it. But I will guess that the research process isn't vastly different between the experience you have and the degree you seek. That is an important asset. That process can be widely applied and tailored to specific areas.