How to know which user update/insert or delete in MYSQL database?
Depends on what version you are. 

You can install the audit_plugin:

INSTALL PLUGIN server_audit SONAME 'server_audit';


You can choose two outputs, file or syslog. The output format will be:

20140901 15:19:44,localhost.localdomain,root,localhost,4,133,WRITE,employees,salaries,
20140901 15:19:44,localhost.localdomain,root,localhost,4,133,QUERY,employees,'DELETE FROM salaries LIMIT 100',0


WARNING NOTICE

If you enable the audit_plugin (it's right with all logging processes like SLOW LOW, GENERAL LOG...) be careful to not saturate your disks in term of I/O. Depends of your workload, prefer write these types of logs on dedicated disks to minimize the impact on your MySQL instances. You can also throw them to syslog but you should have the architecture to handle them (ELK for instance).

Max.