How to use Grep command to identify the particular VS and their state?

grep works on a linefeed boundary, whereas the tmsh output contains multiple lines.

While you can't do this in tmsh since the grep doesn't support -A there, you could use something like this on the command line:

tmsh show ltm virtual|grep ^Ltm:: -A5|grep -e 'Ltm::\|\ \ Availability\|\ \ State'

here's me running that:
[root@snk-5:Active:Standalone] config  tmsh show ltm virtual|grep ^Ltm:: -A5|grep -e '^Ltm::\|^\ \ Availability\|^\ \ State'
Ltm::Virtual Server: next4p81  
  Availability     : unknown   
  State            : enabled   
Ltm::Virtual Server: next4p82  
  Availability     : unknown   
  State            : enabled   
Ltm::Virtual Server: next6p81  
  Availability     : unknown   
  State            : enabled   
Ltm::Virtual Server: next6p82  
  Availability     : unknown   
  State            : enabled   
[root@snk-5:Active:Standalone] config