Matthew 27:3-7

3 Then Judas, which had betrayed him, when he saw that he was condemned, repented himself, and brought again the thirty pieces of silver to the chief priests and elders,
4 Saying, I have sinned in that I have betrayed the innocent blood. And they said, What is that to us? see thou to that.
5 And he cast down the pieces of silver in the temple, and departed, and went and hanged himself.
6 And the chief priests took the silver pieces, and said, It is not lawful for to put them into the treasury, because it is the price of blood.
7 And they took counsel, and bought with them the potter's field, to bury strangers in.

Acts 1:16-18

16 Men and brethren, this scripture must needs have been fulfilled, which the Holy Ghost by the mouth of David spake before concerning Judas, which was guide to them that took Jesus.
17 For he was numbered with us, and had obtained part of this ministry.
18 Now this man purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out.

The assertion is that the Bible teaches Judas died in two separate ways.

In reality, we are simply given two descriptions of what happened, and the chronology is fairly easy to deduce:

1. Judas tried to return the silver
2. He hung himself in a field
3. His rotting corpse hung there until it fell down, his guts spilling out
4. The chief priests bought the field with the silver that Judas tried to return
5. Therefore, Judas "bought the field" with the reward of iniquity

Judas is said to have "bought the field", because his money did so by proxy. In the same way, a rich man leaves an inheritance to a charity, and is said to have "donated" to the cause, even though he's already dead.

Conclusion

Comparing one Gospel with the other simply gives us more detail, and the fact that each account gives a different description of the same event does not constitute a contradiction, at all.