Alexa can't play (and isn't even aware of) music stored locally. All music that Alexa can play is processed on the cloud and streamed to the device, so Alexa won't natively support playing music.
That said, if you're willing to store your music on a cloud hosting service (with HTTPS only1), you can build a skill to do that.
skill-sample-nodejs-audio-player (an official Alexa sample skill) could be adapted to play your local music, by cloning their code and following the steps provided.
This is, unfortunately, very tedious. Alternatively, you can upload up to 250 songs to Amazon Music for free:
To play your personal music from iTunes, Google Play, and more on your Alexa device, you can use Amazon Music for PC and Mac to upload your collection to My Music on Amazon from your computer. After you upload your music, you can ask Alexa to play your music and control playback with simple voice commands.
You can upload up to 250 songs to My Music for free. To upload up to 250,000 songs, you can switch to an Amazon Music storage subscription. To learn more, go to Change Your Amazon Music Storage Subscription.
1 Amazon list the conditions for the hosting service here:
The audio file must be hosted at an Internet-accessible HTTPS endpoint. HTTPS is required, and the domain hosting the files must present a valid, trusted SSL certificate. Self-signed certificates cannot be used. Many content hosting services provide this. For example, you could host your files at a service such as Amazon Simple Storage Service (Amazon S3) (an Amazon Web Services offering).
The supported formats for the audio file include AAC/MP4, MP3, HLS, PLS and M3U. Bitrates: 16kbps to 384 kbps.