Do you wish to improve an existing friendship and/or grow closer to someone you already know? In most cases the answer is relatively simple. However, it isn't always easy because it may involve becoming vulnerable to an extent and moving out of your comfort zone. Of course there are always times to cut up and act stupid for some fun times and good laughs, but in order to grow closer you should make it a point to start a serious conversation with an understanding of mutual respect and without judgment of what the other person may say. This can involve sharing personal truths about yourself, your desires, goals, and beliefs. Of course there is nothing wrong with being a private individual, but being so private in terms of never sharing personal things with anyone or always thinking it is never anyone's business or never wanting anyone to know your secrets or your deep (maybe dark) thoughts can be unhealthy as well as selfish.
The (relatively) simplest answer to how to grow closer to an existing friend is to practice being selfless.
As always you want to find the right balance. You don't want to be too private but you also don't want to over-share; over-sharing, especially too soon in a blooming friendship, can be too personal too fast for some people to handle, and it could come across as the characteristic of being desperate or clingy if you confide in someone too deeply at an inappropriate time which might scare them away from the idea of growing mutually closer with you. You will have to be the judge of this; you should just go with your gut on this and gauge at what level of friendship you have achieved with a particular person and determine how much you can share with that person under the right set of circumstances.
It is a selfless act to share your personal truths and desires rather than keep them to yourself all the time.
Additionally, nurturing a friendship indubitably will go and should go beyond the practice of sharing things about oneself and just talking about things. Acts of service or kindness or grace and forgiveness will allow for a deeper friendship. Being selfless will help you with these acts. Specific examples aren't necessary in this case if you simply think to yourself in situations: "Is this better serving myself or my friend?" The answer will lead you to selfless acts within your friendship, and this kind of sacrificial friendship will be noticed and deeply appreciated by your friend. If this is a friend worth having, then most likely they will be willing to make some sacrifices as well for your sake and happiness and to show some selfless appreciation for the kind of friend you are to them. (If a person or "friend" is unwilling to any extent be gracious to you in any manner despite what you may do for them, then you may suck at picking friends. Picking the right friends to grow closer to is a topic for another time.)
Good luck with improving your friendships and being selfless with others. You may find that putting your selfish ways behind you will bring more fulfilling friendships. May your friends bless you as you bless them.