I am not a native English speaker. Recently I tried to read "A Tale of Two Cities" by Charles Dickens. I gave up soon because I was not able to comprehend that book! I even tried to read the summary of the individual chapters and then read the book. It still didn't help. My question is whether Native-English speakers can read the book "leisurely" without much effort or is it a classical masterpiece which is little difficult to comprehend?
Yes, but it takes some intelligence and considerable education to read it without any effort at all. For example, the opening sentence:
It was the best of times,
it was the worst of times,
it was the age of wisdom,
it was the age of foolishness,
it was the epoch of belief,
it was the epoch of incredulity,
it was the season of Light,
it was the season of Darkness,
it was the spring of hope,
it was the winter of despair,
we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way — in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only.
is mostly poetic license, and relatively comprehensible -- at least, up to "in the superlative degree of comparison only". You'll be happy to know that even native speakers have trouble understanding what this means.
The next part requires some scholarship as well:
There were a king with a large jaw and a queen with a plain face, on the throne of England; there were a king with a large jaw and a queen with a fair face, on the throne of France. In both countries it was clearer than crystal to the lords of the State preserves of loaves and fishes, that things in general were settled for ever.
Dickens' comment about "the loaves and the fishes" alludes to the New Testament, specifically the miracle of the loaves and the fishes. If you don't get this reference, you won't understand the irony. Dickens is comparing Jesus feeding the hungry with the private preserves of wild game kept by the lords of France and England for their pleasure hunting, suggesting that these lords happily continued their sybaritic lifestyles at the expense of the destitute and starving. To these lords, the situation was natural and proper and would continue forever.
And so it continues throughout the book. Many paragraphs reference events or figures that would have been known to Dickens' readers, and which mean nothing to us. These are often unrelated to the main story, and you can usually just read past them if you wish.
You might appreciate using something like SparkNotes while reading, which helps translate the original text into (somewhat oversimplified) modern English. Most students who read "A Tale of Two Cities" do so with some kind of this help.