Individual records:

Most runs scored by a batsman in a home season
In the 2016/17 season, India’s skipper Virat Kohli was in some form with the bat. In the 12 matches, Kohli scored a total of 1,847 runs across formats while out of 1,489 runs were scored as captain (In ODIs against NZ, MS Dhoni was the captain). In terms of the record, Kohli is on top of both the list. In the case of most runs scored across the formats by a batsman in a season, Australian batsman Steven Smith was the previous record holder with 1,647 runs (2014/15).
On the other hand, former Australian captain Ricky Ponting was the previous record holder for most runs scoring in a home season (across format) as captain with 1,473 runs (2005/06).
Most Test runs scored by a batsman in a home season
Apart from Virat Kohli, another Indian batsman Cheteshwar Pujara was also in a very good form in the 2016/17 Test season. Playing all the Tests at home in the season, Pujara scored 1,316 Test runs while Kohli scored 1,252 runs. In the list of most Test runs scored in a home season, Pujara is on top while Kohli ended in the second position. The previous record holder was former Indian batsman Virender Sehwag with 1,105 runs in the home Test season in 2004/05.
On the other side, Cheteshwar Pujara became the second most Test runs scorer in a season after scoring 1,316 runs in 2016/17. Former Australian batsman Ricky Ponting holds the record with 1,483 runs in 2005/06. Gautam Gambhir (1,269 runs in 2008/09) was the previous most Test runs scorer in a season for India.
Most no.of Test double-century scored by a batsman in a season
Indian skipper Virat Kohli scored 3 Test double-centuries in 2016/17. With this no. of Test double centuries, Kohli achieved some records which are as follows:
i) Most no.of Test double-centuries by a batsman in a home season.
ii) Most no.of Test double-centuries by a captain in a season.
iii) Joint most no. of Test double-centuries by a batsman in a season. Former Australian batsman Sir Donald Bradman (1930) and former Indian batsman Rahul Dravid (2003/04) had also hit 3 Test double-centuries once in a season.
Most no. of century partnership by a pair in a home Test season

In 2016/17 home Test season, Indian batsmen Murali Vijay and Cheteshwar Pujara put on 100+ runs partnership, six times. It is the most no.of century partnership by a pair in a home Test season. The previous record was 4 which was made by numerous batting partners.
On the other hand, Vijay-Pujara’s record is also the second most no. of century partnership by a pair in a Test season. The record is seven and the partners were former Australian Matthew Hayden and Ricky Ponting (2005/06).
Most Test wickets by a bowler in a season

Indian off-spinner Ravichandran Ashwin picked up 82 Test wickets in 2016/17. It is the most no. of Test wickets taken by a bowler in a season. South African pacer Dale Steyn was the previous record holder with 78 Test wickets in 2007/08.
Meanwhile, another Indian left-spinner Ravindra Jadeja ended into the 3rd position in this list with picking up 71 Test wickets in the latest season (2016/17).