A simple must-use plugin for WordPress that will provide a `/feed/json` endpoint for converting RSS feeds to JSON.

rss-to-json.php

<?php

/**
 * A simple must-use plugin for WordPress that will provide a `/feed/json` endpoint for converting RSS feeds to JSON.
 * In the absence of any URL parameters, it will convert the main feed to JSON. If a URL is set, it will convert the
 * feed from that URL into JSON. If a callback is set, a JSONP implementation will be returned.
 *
 * Note: Be sure to flush rewrite rules manually by visiting the 'General'->'Permalinks' page in the WordPress admin. If
 * you are using this in a plugin, be sure to flush the rewrite rules on activation using the 'flush_rewrite_rules()' function.
 *
 * @author Micah Wood <micah@wpscholar.com>
 */


add_action( 'init', function () {

	add_rewrite_endpoint( 'feed', EP_ROOT );

} );

add_action( 'template_redirect', function () {

	if ( 'json' === get_query_var( 'feed' ) ) {

		$jsonp = filter_input( INPUT_GET, 'callback', FILTER_SANITIZE_STRING );
		$url = filter_input( INPUT_GET, 'url', FILTER_SANITIZE_URL );

		if ( empty( $jsonp ) && empty( $url ) ) {
			$url = home_url( '/feed/' );
		}

		$request = wp_safe_remote_get( $url );
		$response = array( 'success' => false, 'data' => (object) array() );

		try {

			if ( 200 !== wp_remote_retrieve_response_code( $request ) ) {
				throw new Exception( 'Unable to fetch feed.' );
			}
			$xml = new SimpleXMLElement( wp_remote_retrieve_body( $request ), LIBXML_NOCDATA );
			$response['success'] = true;
			$response['data'] = $xml->channel;

		} catch ( Exception $e ) {

			$response['data']->message = $e->getMessage();

		}

		if ( $jsonp ) {
			@header( 'Content-Type: application/javascript; charset=' . get_option( 'blog_charset' ) );
			echo $jsonp . '(' . wp_json_encode( $response ) . ');';
			die;
		}

		wp_send_json( $response );

	}

} );

Sample usage (same domain): http://{yourDomain}/feed/json/?url={feedUrl}
Sample usage (cross domain): http://{yourDomain}/feed/json/?url={feedUrl}&callback={jsonpCallback}

Sample JSONP implementation:

var url = 'http://{yourDomain}/feed/json/?url={feedUrl}&callback=rss';
var jsonp = document.createElement('script');
jsonp.src = url;
document.body.appendChild(jsonp);

function rss(data) {
    console.log(data);
}

How i can install into my website?