Is there any way to make a custom form template that allows a user to input information via text boxes and drop down lists using aspx. I have a form built, but I'm not sure how to put it on the sharepoint site as a document template. I don't want to use infopath for this as it is meant to replace an infopath form.
That depends on how you build your ASPX page.
If this is one off page - use SharePoint Designer. Create new pages as needed for the library, remove the default markup and replace with your own. You can search for how to do this and get a lot of examples. Once you have your page - you can manipulate the template directly (via XSL) or use the Client Side Object Model to save data back to the list (review the MSDN article on that here)
If you have custom code to deploy you need to use package it as a SharePoint solution. While I know you can do this when you package the entire list I"m not sure about just deploying the form itself.
If you don't need to expose any default list behavior - you could just host the page in a library itself (assuming no code behind) and just place the link to where you need users to enter data in the form.
check out this reference here - it's step by step on what you want to do http://social.technet.microsoft.com/wiki/contents/articles/23955.sharepoint-2013-building-custom-forms.aspx