We are using an external email service, and we are planning to deploy a local Exchange Server 2013 in our location. Question here is, can we continue to use this external provider as our mail connector and forward those emails in that mailboxes to relevant mailboxes in Exchange Server. I have look the similar question related to 2007, but is it still same way in the 2013, or is there any other way to do this in 2013.
Exchange isn't really designed to do that and it can get messy very quickly. What you are talking about though is sharing the SMTP address space.
Take a look at this question for more details. Running company emails at two different suppliers
The version of Exchange is completely immaterial - the same issues apply whether it is two Exchange servers, Exchange and Gmail, Exchange and some random Linux mail server etc. The only difference is how you move mail between the two servers and avoiding email loops.
Personally I would either move everyone in house, or go with something like Office365 (either hybrid with on premise or native). A mixed environment with some on Exchange and some not will cause nothing but headaches, for you as the admin and for the users who will not know whether some are on Exchange and some not.
BTW, i have solved the issue with configuring on-premises exchange server on to a sub-domain. Forwarded mails from service provider to local server and used local server for sending mails. Maybe this solution maybe useful for people in same situation.