When you say “local WP files” are you referring to the WordPress software itself? You can download the current version here at wordpress.org (or even older versions if you need it).
As long as you’re not dealing with dramatically different versions of WP, then the process would be to install your (fresh copy) of WP on your new server, then using either phpMyAdmin or a plugin such as WP-DBmanager (there are many to choose from) you do a ‘restore’ of your backed-up database from your local host (that contains your content, users, etc) into your new WP install – the tables from your old database should overwrite those setup by your new install. If you have errors, post them, but also it’s important when you’re backing up a WP database that you check the box to ‘create (tables) if not exist’ – this makes sure that any tables that *aren’t* setup by WP (such as those added by plugins) get setup when you do your ‘restore’ or import. If you didn’t do this the first time, just make another backup and be sure that’s checked.
The only thing you’ll likely be missing then is the Theme files – you didn’t mention what Theme you were using on your localhost but if it isn’t a custom (or heavily customized) Theme you should be able to download a new copy of the same Theme and activate it.
