1.The boy is coming. 2.He is singing. How can I join the two sentences and make a single sentence?

There are many possibilities to join the two sentences, let's explain according to grammar rules:

a.The boy comes singing.

This would be used as stage direction in a play.

b.The boy is coming while singing.

c.The boy is singing while coming.

These two sentences have different emphases, and neither sounds very good.

d.The boy is coming singing.

This doesn't sound very fluent.

 These are better ways to say this:

The singing boy is coming. 

The boy who is coming is singing.The boy who is singing is coming.

None of the others is grammatically wrong, but they don't sound natural.

Other possibilities are

The boy who is coming is singing.

The boy who is singing is coming.

The boy who is comming is singing.

or

The boy is singing while comming.  

with in my knowledge. Thanks.

My answer:

"The boy is coming, and the boy is singing." 

This sentence joins the two facts regarding the boy in once sentence.  A comma is used to coordinate the clauses and improve clarity. 

the boy is coming and singing

the boy came singing

the is coming while singing

There could be two ways to express the two actions:

The boy, who is coming, is singing.

The singing boy is coming.

THE BOY IS COMING SINGING.

THE BOY IS SINGING AND COMING..

The boy sings

le garcon arrive en chantant

the boy is coming by singing a beautiful song