After several failed attempts, I have managed to strip down and modify the MySQL Connector/Net library enough to get it working on DNXCore50. I would strongly advise against using this in production, but it may work for testing purposes while waiting for an official release from Oracle. Here is the repo.
The following basic scenario works, which was my goal:
string connectionString = "server=127.0.0.1;userid=my_user;password=my_password;database=my_db;default command timeout=10;Connection Timeout=5;";
using (var conn = new MySqlConnection(connectionString))
var cmd = new MySqlCommand(@"select col1, col2 from data where id = @id", conn);
var reader = cmd.ExecuteReader();
col1data = reader.GetString(0);
col2data = reader.GetString(1);