People will believe different things about what it means to be Christ-like. Some will never bother themselves with it one way of the other.
Why would we be want to be like him? Look at our world, planet Earth and the men we have encountered and what man has done with the things that God has made. Is there not something better? I believe so and so do others here probably including you.
God is the Creator. He has a purpose and included in that purpose is a desire to please us, but in order for us to be really pleased we must also please Him. He has given us the perfect model or goal to measure against, which is His Son Jesus. How to attain to it? First Jesus lived his life here as a man of flesh with all our man's temptations, to give us a picture... and also to make it possible for us to attain to that seemingly impossible goal of really becoming like him..
God gave us His Son for that picture, but it is not only painted for us not only in four gospels but throughout the Book we call the Bible.
So far beyond is the picture that we need help even to approach it closely. Remember God's words to Moses:
"And he said, Thou canst not see my face: for there shall no man see me, and live." Exodus 33:20
So we were given the Book to describe him for us, but we need more than the Book alone. We need the "eyes to see" what is really in the Book. Those "eyes" for us is the Holy Ghost in us. Before receiving those eyes we were spiritually blind.
At the start the Holy Ghost allows us to see God through a glass darkly, but eventually if allowed to work in us our vision will allow us to see him "face to face". We are promised something which was denied to Moses in the verse I quoted above.
Why would we want to be like him? So we can see his face. What is that we are to be seeing? How about the glory of God?
"When thou saidst, Seek ye my face; my heart said unto thee, Thy face, LORD, will I seek." Psalm 27:8
If we have ever fallen in love with person naturally, we will have a clue as to why we would want to seek his face and to be like him. Love [=God] is as He is. Human love is often in books and poetry and movies played as the ultimate goal for mankind, as if there was nothing higher. I am not saying it not high, but it is a long way from being the highest. Human love in its best example is still only a type or shadow of the real thing. The real thing is the "why" if we can even begin to understand it and to see it:
"For my thoughts are not your thoughts, neither are your ways my ways, saith the LORD.
For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts." Isaiah 55:8-9
The "face to face" should be our goal once we have received our first "through a glass darkly" vision of Him! Remember what Solomon wrote:
"Where there is no vision, the people perish..." Prov 29:18