Is it possible to configure log-in/access logging (success & fail) locally on ASA?
If it's possible how?
If it's not then should I use asdm or TACACS?
Cisco ASA can store logs locally but you will have much better results sending logs to an external syslog server.
It can also use external authentication like radius or tacacs+ but it can store events for local authentication if you want. Take a look at these log events for authentication and command logging:
Those should help you get started logging what you want but I would strongly recommend storing logs on a syslog server. Storing the logs in memory runs out quickly and you won't find what you want a lot of the time when you need to go back and look for it.