PhD after completing MSc in Botany

Not required. You can do a PhD either in Biochemistry or Genetics. I hope you have studied Chemistry at your graduation. But getting a good place in a good institute is the key. Many Professors prefer the people who have PG in the concerned subject. So the key is in getting the approval of a good guide for your work.
If you know any Professor in any of the foreign University in the subjects you have mentioned, you can contact him and ask for his opinion and get his approval to be your PhD director.
If you don't know anybody, you make a beautiful resume of yours giving the actual details and highlighting any special points you have like a Gold medal, Scholarship, awards etc. Make a small note on the subject on which you want to conduct your research can be made and attached to your resume. Please send this to any professor who works in a similar area in the various institutions where you want to join for a PhD. Once he goes through your mail if he likes he may respond.
Another option is to talk to the Professors at the university where you did your PG and get the references to them and apply to those people attaching the testimonials from your professors. If possible make your professor send a mail to his connection asking to help you in this regard.
All the best to you.

Hi,
PhD. is a research-based degree which covers a vast area of any subject. In your case, you want to pursue in the field of Biochemistry or Genetics but you have specialization in Botany.
You can pursue your Ph.D. in any topic from botany which is connected to its genetics or Biotechnology. Here I mean to say that Botany also has Biochemistry and genetics. For example, you can take any topic related to photosynthesis which will include Biotechnology and any topic related to the growth of any particular plant with variations will include genetics.
In the field of Biotechnology, all the subjects are interrelated and you can choose the subject and topic accordingly. The important thing to consider is your interest and knowledge because it is really difficult to work for 5-6 years on the subject in which you do not have an interest. Another point is the availability of topics and guides as per your choice.
I am from the field of biotechnology and worked in Bhabha Atomic Research Centre, Mumbai. In my lab, there was one trainee pursuing M.Sc. in Botany who after M.Sc. registered for Ph.D. under the same guide and selected the topic of Histone modifications in cereals. This was a combination of botany and molecular biology.
So, search for interesting topics, guides, etc and then register.
All the best!

A Ph.D in the field of other than Botany ie in Biochemistry or Genetics can be taken up provided your guide finds you fit to take such perusal with the close interaction in a interview. He will assess your grip and ability to understand the topic assigned by him for the Reasearch.
In order that get success in the assigned topic, you need the have a talented guide offering you all sorts of help in the alloted area where you are sustaining your activities apart from a good library containing books offering you materials for easing your Research- activities.
However, you have to trace out an university of repute in addition to a capable guide so that you may proceed in your area of Research smoothly.
Your own area from where you have done your post - graduation can be of immense help if you interact with the professors in the field of Biochemistry or Genetics.
You can also refer to Research - journals where you may get published articles of the scholars under the guidance of the foreign guide. You may contact such professors and show your willingness to carry out Research - work under his guidance.
You may get a positive help after offering him your detailed resumes providing him all the relevant facts.

It may vary from university to university. As far as I know, you can easily get the opportunity to enroll yourself for a Ph.D. in Genetics, more specifically in Plant Genetics. But, Biochemistry may be a little bit difficult. Because, for enrolling in Ph.D. in a topic related to Biochemistry, first preference will be given to the students of Biochemistry.

I feel you will get the opportunity in Botany or in Plant Genetics.

If you have an intention of doing PhD in biochemistry or genetics from abroad then you will have to apply for admission in some university there for this and only after getting admission you can go there to pursue it. You will have to apply in these foreign universities online and in case they ask you to go through some eligibility course there before registering for PhD then you have to take a call on that. I hope you have sufficient financial resources to go abroad or you are managing some loan from the banks for these studies. Of course the foreign students are allowed some fixed hours of working in many foreign countries and you can keep that also in mind as a source of earning there for meeting a part of your educational expenses.

Some of the universities where one can try for admission are - University of Glasgow, UK; The Australian National University, Australia; Boston University, US; La Trobe, Australia etc.