You'll need to pass alpha as true to the renderer:
renderer = new THREE.CanvasRenderer({ alpha: true });
Then you can set it to transparent:
renderer.setClearColor( 0x000000, 0 );
Also added a black background color to the div so we can see it working
.waves { 
  height: 100vh;
  background-color: black
}
/* 
ThreeJs custom waves
Original script by ThreeJS : https://threejs.org/examples/canvas_particles_waves.html
Modified version for Cloudoru by Kevin Rajaram : http://kevinrajaram.com
Date: 08/14/2014
*/
var SEPARATION = 40, AMOUNTX = 130, AMOUNTY = 35;
var container;
var camera, scene, renderer;
/*
if (window.WebGLRenderingContext){
    renderer = new THREE.WebGLRenderer({ alpha: true, antialias: true });
    }
else {
    renderer = new THREE.CanvasRenderer();
    }
*/
var particles, particle, count = 0;
var windowHalfX = window.innerWidth / 2;
var windowHalfY = window.innerHeight / 2;
init();
animate();
function init() {
    container = document.createElement( 'div' );
    document.body.appendChild( container );
    if(container) {
        container.className += container.className ? ' waves' : 'waves';
    }
    camera = new THREE.PerspectiveCamera( 120, window.innerWidth / window.innerHeight, 1, 10000 );
    camera.position.y = 150; //changes how far back you can see i.e the particles towards horizon
    camera.position.z = 300; //This is how close or far the particles are seen
    
    camera.rotation.x = 0.35;
    
    scene = new THREE.Scene();
    particles = new Array();
    var PI2 = Math.PI * 2;
    var material = new THREE.SpriteCanvasMaterial( {
        color: 0xFFFFFF, //changes color of particles
        program: function ( context ) {
            context.beginPath();
            context.arc( 0, 0, 0.1, 0, PI2, true );
            context.fill();
        }
    } );
    var i = 0;
    for ( var ix = 0; ix < AMOUNTX; ix ++ ) {
        for ( var iy = 0; iy < AMOUNTY; iy ++ ) {
            particle = particles[ i ++ ] = new THREE.Sprite( material );
            particle.position.x = ix * SEPARATION - ( ( AMOUNTX * SEPARATION ) / 2 );
            particle.position.z = iy * SEPARATION - ( ( AMOUNTY * SEPARATION ) - 10 );
            scene.add( particle );
        }
    }
    renderer = new THREE.CanvasRenderer({ alpha: true });
    renderer.setSize( window.innerWidth, window.innerHeight );
    
  renderer.setClearColor( 0x000000, 0 );
    container.appendChild( renderer.domElement );
    window.addEventListener( 'resize', onWindowResize, false );
}
function onWindowResize() {
    windowHalfX = window.innerWidth / 2;
    windowHalfY = window.innerHeight / 2;
    camera.aspect = window.innerWidth / window.innerHeight;
    camera.updateProjectionMatrix();
    renderer.setSize( window.innerWidth, window.innerHeight );
}
function animate() {
    requestAnimationFrame( animate );
    render();
}
function render() {
    var i = 0;
    for ( var ix = 0; ix < AMOUNTX; ix ++ ) {
        for ( var iy = 0; iy < AMOUNTY; iy ++ ) {
            particle = particles[ i++ ];
            particle.position.y = ( Math.sin( ( ix + count ) * 0.5 ) * 20 ) + ( Math.sin( ( iy + count ) * 0.5 ) * 20 );
            particle.scale.x = particle.scale.y = ( Math.sin( ( ix + count ) * 0.3 ) + 2 ) * 4 + ( Math.sin( ( iy + count ) * 0.5 ) + 1 ) * 4;
        }
    }
    renderer.render( scene, camera );
    // This increases or decreases speed
    count += 0.2;
}
canvas{
  position: absolute;
  top:0;
  left:0;
  z-index:2;
}
.waves { 
  height: 100vh;
  background-color: black
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/r68/three.min.js"></script>