Below is one perspective based on a Gaudiya scripture and mostly my own explanation based on some experience in regular chanting, under guidance of some senior teachers.
Generally Sraddha is the word used for faith and in it's highest degree, Lord Caitanya equates it with dhrda Visvasa (firm conviction) in Krishna.
Sri Caitanya Mahaprabhu explains Sraddha as follows:
CC Madhya 22.62 — “Sraddha is confident, firm faith that by rendering
transcendental loving service to Krsna one automatically performs all
subsidiary activities. Such faith is favorable to the discharge of
In terms of chanting, we hear from sadhu, sastra about the importance of holy names. We may have never chanted before but we read from scriptures, see the sadhus chanting, find out about the practical impact on people's lives and start chanting. This is a demonstration of faith. As you chant more, based on your personal experience and scriptural knowledge your faith increases. Faith in its highest stage is firmly beleiving and execute with firm conviction as cited by Caitanya Mahaprabhu above.
A simple dictionary meaning is "free from pretense or deceit; proceeding from genuine feelings". Sometimes this is also used to mean how comitted we are to the practice. Say I make a commitment to chant, I claim I have faith that this is the most important thing but I always keep it on the low priority of my 'to do' list. I get up, read news, play, go to office and when there is nothing else to do I chant. This means I am not sincere, I made a pledge of faith that it is most important but it is not genuine. If it was most important I would have done it at the first available opportunity, i.e, in the morning before doing other tasks and not kept it to the end of the day.
Process of chanting:
Chant at earliest possible time, like in the morning before sunrise. First we focus on hearing the syllables of the mantra distinctly. We chant little loudly so that we can hear the sound of our own chanting. This way our tongue and ears are both engaged in the holy name. The holy name is very powerful and as it enters our heart through our ears it can eventually reveal the form and pastimes of the Lord. It is said the name will eventually reveal the form and pastimes of the Lord. We do not force our selves to think of the form but just focus on hearing. We can chant in front of the deities of Radha-Krishna, Sita-Rama, Laksmi-Narayana etc. so our eyes are also engaged in seeing Their beautiful forms.
Also, what is chanting with faith for a person who expects nothing in return from the mantra?
I don't fully understand your question but it seems you are asking about the faith in the highest stage. A person who attained highest faith in the name of the Lord doesn't see the Lord and His name to be different. They are not afraid of any impediments put in their chanting. Faith of Haridas Thakura (an associate of Caitanya Maha Prabhu) is an example of this. He was beaten to death but he didn't give up chanting and the Lord saved him. You can read the story here We also see this in the case of Prahlada, who was threatened by his demoniac father Hiranya Kasipu but the body didn't give up His devotion to the Lord. Lord Hanuman is also another good example who has firm faith in the name of Lord Rama.