How does a computer compute negative(-) and positive(+) Infinity?
The simple answer is that a computer that follows a floating point standard such as IEEE 754 does not really compute +/- infinity at all. Instead it treats it as an exception - a special case. Whenever the processor detects an attempt to divide a non-zero floating point number by zero, it sets the result to a special value that is then interpreted as +/-infinity, where the sign depends on the sign of the original number.

In IEEE 754 I believe the special values of +/- infinity are represented internally as a floating point number with a sign bit, an exponent set to all $1$s, and a significand set to all $0$s. This is a value that cannot result from normal (non-exceptional) floating point operations.