The best way to make your boyfriend see that the lack of sex is killing you is to speak directly. You can either tell him outright by calmly describing what he is doing--or not doing--and how you feel about it. For example, 'I am really upset over not having regular sex. 'Or you can ask him a question to spark his awareness regarding the impact that he's having on you. For example, 'Are you aware of how upsetting the lack of sex is for me?'
It sounds to me as though you have already told him how you feel. His excuses as to why he's not stepping up to the plate must have been delivered in response to your complaints over his lack of service. This is why I suspect that there is more to the lack of sex than meets the eye. If I had to surmise, I would say that your boyfriend has a passive-aggressive personality disorder, which means that he expresses his angry feelings by withholding what you want. If I am right, then he has never learned how to directly communicate his negative feelings (most likely because his parents punished him whenever he tried to speak his anger); and he learned to bury his anger at a young age.
Since anger can't remain buried forever, it necessarily will resurface in all kinds of indirect ways. Soon, indirect expression of anger becomes the normal way of handling negative emotions and this mode of operating becomes embedded in the character structure. This means that changing the way he handles aggression will be difficult. It can be done, but the person must be motivated to change.
Before we can find out if he's motivated to change, we need to know if he's even aware of the problem that he has. Keep in mind that helping him become aware of the problem is difficult in and of itself, since that requires his owning the fact that he isn't owning his anger. Since he is afraid to admit his anger, he will be reluctant to own that he isn't owning it, if you see what I mean. If you decide that you still want to take on this mission, then you can take one of two approaches.
You can either help him become aware of the angry thoughts and feelings that are behind his refusal of sex; or you can try to help him to realize how his withholding behavior makes you feel. Realizing how hurt and angry you are may help him realize that he wouldn't be acting in ways that leave you with such feelings unless he were angry himself.
To achieve the first goal--to get him to become self-aware--you could ask, 'Do you know how you are feeling toward me when you avoid sex for such a long time?' or 'Do you know what you are trying to tell me when you avoid sex?' To achieve the second goal--to get him to realize how his behavior makes you feel--you could say, 'How do you want me to feel about the lack of sex?' or 'What message are you sending me when you don't want to have sex with me?' or 'Do you want me to feel hurt (rejected, angry, or whatever you do feel) over the lack of sex?' or 'Why would you want to behave in a way that pushes me away? Are you angry with me?' or 'Why would you behave in a way that make me so angry? Are you angry with me?'
If you can get him to own his feeling, then you need to help him to see that the way he expresses his anger (by withholding) is destroying the relationship. See if he is motivated at this point to go for therapy and work on this pattern. If he is willing to work on this issue, you will need to make sure that you check in with him often, inviting him to communicate his angry feelings before they build up and then resurface in a sneak attack.
Your encouraging him to talk will counteract his parents' stiffling of his anger, and in time he will feel safe to speak up on his own without your prodding. If you can't get him to open up and own the angry feelings that are fueling his withholding behavior, then make an appointment with a couples therapist and put his foot to the fire in front of a neutral party.