The best way involves installing the kodi-alexa skill in conjunction with kodi installed on a low power device such as a raspberry pi. Unfortunately the skill is not available from Amazon directly but the install isn’t too difficult so long as you are fairly confident with computers.
So long as you add a music folder and subsequently ‘scan items to library’ in kodi (which catalogues your collection), you can access as many MP3 files you want (you are only limited by the size of storage you have connected to your raspberry pi).
This approach allows you to ask things such as ‘alexa, ask kodi to play songs by the Beatles’ or ‘alexa ask kodi to play paradise by Coldplay’. In addition, if you connect your raspberry pi to a tv, you can also ask alexa to play a particular film or to series available locally on kodi.
The only annoyances I have with this approach are (1) the questions you need to ask are unnecessarily long but i think this is common for any alexa third party skill (Ie it would be easier to simply say "kodi, play paradise by Coldplay’ rather than “alexa ask kodi to play Coldplay”) and (2) it takes a few seconds for the kodi box to search for the music/video you want. However I suspect the search function offered by this skill will improve over time since it is still being developed and improved by a relatively large number of developers.
Step 1 of installing kodi on a raspberry pi is easy simply install the openelec operating system on a raspberry pi. Granted, if you don’t have one of these already then you are perhaps better off paying Amazon the £25 per year to host your music collection since a raspberry pi is about the same price.
Step 2 (setting up kodi so it accessible to the outside world, uploading your very own kodi alexa skill) is a bit more involved but there are a few good guides available.
I would suggest reading the following first which explains how to create user ids with lambda and Amazon for developer access:, as well as the stall f the skill itself:
If you fail still, then I refer you to posts 1102 and 1097 from the following forum thread which literally give you a step by step guide assuming you have already set up the relevant developer profiles with lambda aws etc.