This is a question that we commonly receive. I think that an answer to it, although there are many, can be found in the example of the Prophet Muhammad (pbuh). There are three elements constantly present in the life of the Messenger that can be adapted and applied to our own lives as well. If present they can help us to deepen our faith and improve our conduct.
1) Anchoring Oneself in the Revelation
We cannot deepen our faith if we do not understand it. We cannot improve our character and conduct if we do not know what we should be striving towards. The first element in the whole process is to anchor oneself in the revelation.
This is clear in the example of the Prophet (pbuh) since he was the relayer and clarifier of the revelation itself. In doing so he was in constant contact with the revelation and the revelation was guiding and influencing everything that he did.
For us this means having a constant connection with the revelation by reading the Quran regularly with understanding and learning about the life and behavior of the Messenger. This can also take the form of regular reading of books of tazkiyah, which, although they are not scripture themselves, are based on scripture. In fact, I think it is good practice for every Muslim to be constantly reading some book of tazkiyah. Each book can be read many times since the lessons never get old and the goal is reminding ourselves of the practices that we need to embody. A good place to start is here.
2) Being in the Company of Those Who are in an Elevated State
If we are honest with ourselves and truly want to grow, we have to acknowledge that some people (to the best of our knowledge) are at a more developed state in their faith and character than we are. This understanding leads to one of the most central concepts in Islam, suhba. The word literally means companionship and is also the source of the word commonly used to refer to the Companions of the Prophet Muhammad (pbuh). The idea is to find people that one can learn from, in manners, conduct, and spiritual experience, and spend time with them. There are things that are learned through close companionship that cannot be learned in books or anywhere else.
In the life of the Prophet (pbuh) you see him acting upon this in his relationship with the Angel Jibril (as). Jibril comes at different points in his life and teaches him certain things, he corrects him on my mistakes, and they had all kinds of interactions. It was an interestingly close relationship, but also a relationship wherein the Prophet was always under the watchful eye of an instructor and guide. We also see it in the way his companions interacted with him. They were very keen to stay near him and learn from the details of his experience as much as they possibly could.
For us this means seeking those who we can learn from. By this I do not mean to only go to people of knowledge. In this case we are specifically looking for people who have a deep consciousness of Allah and beautiful character. These two go hand in hand because character is related to concern and attention to others and the person who is high conscious of Allah will also be highly conscious of the people. You should be able to feel the impact of their faith on their life and their character and manners should inspire you to be better.
3) Being in the Service of Others
A great way to draw closer to Allah and improve ourselves is to be in the service of others. The Prophet (pbuh) said, "God is in the aid of his servant as long as His servant is in the aid of his/her brother/sister." This is very simple and clear. Serve others, humbly and with gratitude for the opportunity, and God will be in your aid.
This is exemplified in the life of the Prophet (pbuh) by everything that he did. He was constantly concerned with and carrying for his companions and people in general. In one hadith he gives a beautiful parable of this relationship. He said (paraphrased), "The example of me and you is like the man who makes a fire. When he does so all the flies try to throw themselves into it and he is doing all that he can to stop them." The Prophet is doing whatever he can to lead people to guidance and salvation, this is the highest level of concern. However, he did not stop at guidance, but also served the people who were most in need. When the revelation began his wife encourage him saying that God would not forsake him. Among the reasons that she mentioned was that he supports the weak and gives to the needy.
In our own lives this means that we should be concerned with the problems of others and want to help fix them. This can occur on a micro level through day to day kindnesses and support to those who are in need and it can occur on a macro level through advocacy and policy formation to improve structures that can negatively or positively affect the lives of thousands and thousands of people.
The question "How can I improve my faith and character" is an important one. It can be answered through a look at the life of the Prophet Muhammad (pbuh). In doing so we find three major elements that were consistent in his life and when they are adapted to our experience they can be greatly beneficial to us. They are: 1) Anchoring oneself in the revelation; 2) Being in the company of those who are in an elevated state; 3) Being in the service of others.