Your example sentence
Make sure you spread the ointment into the skin
is grammatically correct and understandable with the meaning
Apply the ointment so that it penetrates the skin
however the phrasing may be found awkward by native speakers. The awkwardness arises from the use of spread...into.
Skin is a surface which things are usually placed on, over, or through and sometimes into
The wind on his skin made him feel cold
She poured water over her skin to wash it
The splinter pierced through his skin
The injection was made into the second layer of skin
Ointment can be applied over skin and then allowed to penetrate into the skin due to the latter's permeability.
Make sure to spread the ointment over your skin until it is absorbed
may convey the message you want with less awkwardness