If it's supposed to edit its own post only, make it with author's capabilities. If it should edit posts of other users, as well, make it with editor's capabilities.

Case 1

function wpse_add_your_role() {
    $role = get_role( 'author' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Case 2

function wpse_add_your_role() {
    $role = get_role( 'editor' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Code location

In both cases, put the code in current theme's functions.php file. Once the new role is created, remove the code from functions.php, or comment out this line:

// add_action( 'init', 'wpse_add_your_role', 10 );

If it's supposed to edit its own post only, make it with author's capabilities. If it should edit posts of other users, as well, make it with editor's capabilities.

Case 1

function wpse_add_your_role() {
    $role = get_role( 'author' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Case 2

function wpse_add_your_role() {
    $role = get_role( 'editor' );
    if ( ! empty( get_role( 'your_role' ) ) ) { 
        remove_role( 'your_role' );
    }
    add_role( 'your_role', 'Your Role', $role->capabilities );
}
add_action( 'init', 'wpse_add_your_role', 10 );

Code location

In both cases, put the code in current theme's functions.php file. Once the new role is created, remove the code from functions.php, or comment out this line:

// add_action( 'init', 'wpse_add_your_role', 10 );

Final notes

You can fine tune your role, using $role->add_cap() or $role->remove_cap() methods, according to your needs. In case 1, do not remove edit_posts capability. In case 2, do not remove, neither, edit_posts nor edit_others_posts capabilities.