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So I want to use hibernate search to index it in my personal computer THAN move it to my remote computer site.
It will take some time to index a huge amount of data, you'll have to have a powerful personal computer.
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When I index my data creating the index directory and storing the data in my local database, is there a way to move ONLY the index files to the remote site and then start any (remote) application that would load the index information to the (remote) database?
Index is not stored in the database, you can just copy over the produced files to another system, just make sure that the databases are equal. There is no need to index the data while storing stuff in your local database, you can restore a dump locally and then use index() API in Hibernate Search to rebuild the index only. See the mass indexing chapters for hints and finetuning.
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Or should I move a database backup from my personal computer to the remote site and then create the index there?
You remote computer will need the database as well (In most scenarios, and otherwise there's no point in using Hibernate Search as you could be using just Lucene as well), and having it rebuild the index for you is often preferred: you're sure the db is the same and you have less data to transfer.
You might want a local copy of the index for debugging, but most of the time for debugging it's good enough to have a subset of all the data (and relative subset of the index).