What happens to the task when it reaches the Deadline?

There is nothing really happening. No event will be triggered. But, in a task report it can be shown as overdue. Look for task metrics in the report column configuration. To make something happening you will need to add an escalation or exception.

In that case I have a scenario.

I have kept deadline of 60 minutes for a task. In order to not show Overdue in Task report. I am raising an escalation of timer 15 mins and re- assigning task to X person and for the same I try to raise a level 2 escalation of 30 mins and re-assign to person Y and so on.

In this way I am trying make process as On-time task.  Is the above scenario right as part of handling the deadlines on regular basis? 

In that case I have a scenario.

I have kept deadline of 60 minutes for a task. In order to not show Overdue in Task report. I am raising an escalation of timer 15 mins and re- assigning task to X person and for the same I try to raise a level 2 escalation of 30 mins and re-assign to person Y and so on.

In this way I am trying make process as On-time task.  Is the above scenario right as part of handling the deadlines on regular basis? 

If this fits your requirements, then it should work. If you do not want to use/show the deadline, then just leave it undefined. In a task report it is up to you whether to show the deadline or not.

Task Escalations are there to help 'manage' Tasks to completion but since a User Input Task always relies on a User to complete them then you cannot 'force' them to conform to a Deadline, just remind/encourage/report.