William Blake wrote two corresponding sets of verses called "Songs of Innocence" and "Songs of Experience." In "Songs of Innocence," Blake looked on life as if through the eyes of a child; he presented an innocent, trusting, and optimistic viewpoint on his subjects. In "Songs of Experience," on the other hand, he presented the same or similar topics from a mature, pessimistic, and even dark perspective. In this way, by showing two contrasting sides to a topic, he believed the real truth could be better perceived. 

"The Tyger" is a Song of Experience; its corresponding poem is "The Lamb." "The Lamb" presents God as the Lamb of God, an image that is sweet and comforting; the poem is written as if a little child were addressing a lamb. "The Tyger" is best understood in comparison with "The Lamb." In "The Tyger," God is presented as something like a powerful mad scientist, forging in his heavenly "furnace" a beast ruthless and deadly. The Tyger is a symbol of violence, oppression, and fear. The multiple questions to the Tyger about who made it leave the impression that the speaker is blaming God for the evil and pain that exists in the world and questioning the motives of such a Creator. 

It would be a mistake to say that Blake's purpose in writing "The Tyger" was to show that God is the source of pain and violence in the world, just as it would be a mistake to assume that Blake's purpose in writing "The Lamb" was to convert people to a belief in Jesus Christ. Blake's purpose in writing the TWO poems was to show the contrasting sides of God in hopes of developing a fuller understanding of who God really is.

What is William Blake's point in writing the poem "The Tyger"?

Trying to discuss Blake's "point" in writing "The Tyger" requires that we understand Blake's intention in composing the poem, and we can only guess at a poet's intention--unless we have his own statement about why he wrote a particular poem.  But we can, based on what we know of Blake from his works, try to understand the meaning of the poem.

"The Tyger," which is part of a group of poems called Songs of Experience (1794), is a counterpoint to an earlier collection of poems, Songs of Innocence (1789), in which we find "The Lamb."  In both poems, Blake is asking the same question--that is, "What kind of God made you the way you are?"  In "The Lamb," for example, Blake indicates that the Lamb's creator is "called by thy name,/For he calls himself a Lamb" (ll. 13-15).  This is clearly a reference to Christ, who is often referred to as "the lamb of God."  The same question in "The Tyger" elicits the question, "What immortal hand or eye/Dare frame thy fearful symmetry?"

Both poems question who the creator is, but in "The Tyger," the creator is no longer assumed to be a Christ-like figure who is capable of creating something as gentle as a lamb.  Rather, the tiger's creator is much more powerful:

And what shoulder, and what art,/Could twist the sinews of thy heart? . . . In what furnace was the brain?/What the anvil? what dread grasp/Dare its deadly terrors clasp? (ll. 9-16)

Blake is essentially describing a god who forges the tiger, almost as if the tiger were a mechanism, rather than one who "gave thee [the lamb] life and bid thee feed." (l. 3 of "The Lamb")  The god who created the lamb--described as meek and mild--has become a strong, fierce god who, in his "dread grasp," is capable of creating something that reflects his own strength and fierceness.  The god who makes the tiger is no longer a gentle, loving god, and breathes life into his creation, but a god who creates with a hammer and chain and must use his "dread grasp" to make such a fearsome creature as the tiger. 

By implication, then, Blake is saying that the god who creates something as fierce as a tiger must be even fiercer than his creation, no longer capable of creating innocent things like lambs.    By the time Blake wrote Songs of Experience, he was convinced that English society--from the church to the government to the individual--was falling apart, and this sense of impending doom informs "The Tyger."  His poetic vision of the dissolution of society is "The Tyger" and not "The Lamb."