Can I do Part III of the Mathematics tripos as a part of my PhD in Cambridge University? I am physics master's student in India. I wish to do my PhD in theoretical physics in the Applied Maths department at Cambridge (DAMTP). Can I apply for a PhD and do Part III as a part of it (akin to 1 year of coursework before PhD in India or Master's + PhD in US) or should I do Part III and apply for a PhD in DAMTP separately by the end of it?

Even though I have good grades in all my courses, I am not confident with my skills. My institute is one of the very best in India, however I am not satisfied with my understanding of most of the subjects (and mathematical rigour in general) and there are a few topics, e.g. Advanced QFT, Advanced GR/Cosmology, which are not offered here. I believe Part III would help me in fixing these holes and I want do this before my PhD (in Cambridge or elsewhere).   

From the Cambridge website: "Part III is a 9 month taught masters course in mathematics.  It is an excellent preparation for mathematical research and it is also a valuable course in mathematics and in its applications for those who want further training before taking posts in industry, teaching, or research establishments."

See also the Cambridge website and Wikipedia.
Part III of the natural sciences tripos is essentially a Master's degree: you take courses and exams and write an extended essay. The application process is completely separate to that for DAMPT PhDs (or PhDs in any other of the Cambridge physics departments). 

The simplest option is to do Part III and then apply for PhDs while you are doing it. It's highly unlikely you will have time (and I'm not sure it's even allowed) for you to take Part III courses while registered for a PhD. You could contact someone in Part III admissions to ask them whether you can take courses alongside a PhD, but I suspect the answer would be no.

Personally, I wouldn't place too much weight on the importance of Part III. It has its fearsome reputation for a reason and many people I know who did it, hated it. If you go straight into a PhD, you will learn the relevant knowledge as you go along -- it's unlikely you'll need an in-depth grasp of every part of QFT, general relativity, group theory etc.  Anecdotally, of the other PhDs in my year group (not Cambridge but a similarly reputable cosmology institute), only one out of seven of us has Part III. You easily can succeed in a theoretical physics PhD without it.