Bar Council of India (BCI) does not recognise any LLB degree obtained in distance mode or by correspondence. So even if someone acquires it then it is only for the academic purposes and not for professional practicing as a lawyer. Some people do it to increase their knowledge in the field of law due to certain career reasons or due to carry out some legal works in private mode.
So there is not a single institute which is recognised or approved by the Bar Council in this regard. If in your case it is intended just for knowledge and skill development in the field of law then there are many institutions which offer it in correspondence mode and some of them are – National Law School of India University Distance Education Department, Bangalore; Guru Nanak Dev University, Amritsar; Banaras Hindu University, Varanasi; Kalinga University, Bhubaneswar etc.
Interestingly, if a person is only interested in learning Law and formal degree is not required then there are some specific coaching or tutorial centres which even provide law education to the interested persons. One can explore that option also.
You can pursue your LL.B. course along with your job,provided you take a written permission from your employer. Since there is no provision of evening classes in any law college, hence official permission to attend day classes or appearing in the examinations of the law course is required.
Reference: Bar Council of India