What is a devotee of Krishna supposed to feel while chanting Hare Krishna?
Many devotees have asked this same question of Srila Prabhupada and other senior devotees. At the beginning of our chanting, we have a sincere desire to explore the chanting. We begin to chant on the inspiration of the spiritual master and other devotees and we are deeply inclined to "do it right."
This fresh, simple mentality usually offers some "taste" or inclination to chant carefully. We feel focused, enlivened, and concerned with chanting carefully and attentively. However, at the beginning stages of sadhana (spiritual practice) we mostly chant out of duty, on the order of guru, and whatever taste we experience is secondary. We have material desires and we're not really convinced chanting alone will make us happy. We chant in the morning but throughout the day our mind is attracted to millions of other things.
As we progress, often we become more causal about the chanting. We expect that very quickly we'll see Krishna everywhere. When that's not the case, we become discouraged or mechanical about the chanting. It's difficult to focus the mind, yet we are compelled by the order of guru and the deep knowledge that chanting is what we have to do to purify ourselves and eventually get to know Krishna.
As we progress further, and the heart becomes more pure, we begin to become free from unwanted things within the heart. Our chanting becomes deeper and our longing to understand Krishna through His name becomes more pure and sincere. Our desires for material things decrease. We become more focused. We begin to realize that the Holy Name has everything that we want, and that true happiness can only come from chanting Hare Krishna!
From this point, we truly begin in earnest to chant the Holy Name. More and more, we begin to relish the chanting itself. We actually begin to develop a deep attraction for the holy Name that compels us to continue chanting.
I suggest that you read Nectar of Instruction written originally by Srila Rupa Goswami and translated by Srila Prabhupada. This will help you understand the process further.
I hope this is helpful.