Ok, so it is a very common complaint with cleansers that they do not remove the eye makeup very effectively. Yeah, it really happens. So, I'll share what I do for that. Not that I apply eye makeup but removing sunscreen around the eyes or kajal also requires the same effort.
I use a mixture of almond oil and castor oil. Remember the Oil Cleansing Method in which castor oil is the main ingredient. Castor Oil has cleansing properties apart from its many beneficial effects so it is one of the major oils used for removing residue from the skin. But, using castor oil alone is not recommended for the skin as it is quiet thick and concentrated. So, it is better to mix it with a carrier oil. The reason we choose almond is that it is also effective for removing makeup and aids the growth of eyelashes. So, mix almond oil and castor oil in 3:1 ratio and store it in a glass bottle.
To remove makeup, take a generous amount of the oil and massage the entire area around the eyes. Make sure you have enough oil that your skin glides along the skin coz you don't want to pull and tug at the skin around the eyes which is very thin and can lead to wrinkles and fine lines if disturbed too much. Massage for 5 minutes and remove the oil with cotton very gently without pulling on the skin. You can use face wipes too. And, then, wash off with a cleanser.
Caution: Do not use Coconut Oil as it makes the eyes sting badly. Almond oil or castor oil do not have any such reactions to the eyes.