I would suggest using Polyline functionality from mapbox api, you need to provide a set of points (latitude,longitude) that constitute your route, here is some snippet of code that may help you from mapbox, you can see reference in mapbox api website
Points array for the route
// Create array of lat,lon points var line_points = [ [38.893596444352134, -77.0381498336792], [38.89337933372204, -77.03792452812195], [38.89316222242831, -77.03761339187622], [38.893028615148424, -77.03731298446655], [38.892920059048464, -77.03691601753235], [38.892903358095296, -77.03637957572937], [38.89301191422077, -77.03592896461487], [38.89316222242831, -77.03549981117249], [38.89340438498248, -77.03514575958252], [38.893596444352134, -77.0349633693695] ];
Choose the color you want
// Define polyline options // http://leafletjs.com/reference.html#polyline var polyline_options = { color: '#000' };
Draw the route
// Defining a polygon here instead of a polyline will connect the // endpoints and fill the path. // http://leafletjs.com/reference.html#polygon var polyline = L.polyline(line_points, polyline_options).addTo(map);
Good luck.