Add this to your theme's function.php to show first name and last name fields on woocommerce's registration page.

gistfile1.php

function woo_new_fields() {
?>
    	<p class="form-row">
				<label for="first_name"><?php _e( 'First Name', 'woocommerce' ); ?> <span class="required">*</span></label>
				<input type="text" class="input-text" name="first_name" id="first_name" value="<?php if (isset($_POST['first_name'])) echo esc_attr($_POST['first_name']); ?>" />
			</p>
			<p class="form-row">
				<label for="last_name"><?php _e( 'Last Name', 'woocommerce' ); ?> <span class="required">*</span></label>
				<input type="text" class="input-text" name="last_name" id="last_name" value="<?php if (isset($_POST['last_name'])) echo esc_attr($_POST['last_name']); ?>" />
			</p>
				

<?php
}

add_action( 'register_form', 'woo_new_fields', 10, 2);