SN74LS393N Integrated Circuit is confusing me
The current specification in a datasheet specifies the amount that the part draws from the power supply, not the amount of current that needs to be provided.  When the voltage at the Vcc pin is 5V, the chip will draw the amount of current that it needs, in your case 38ma to 64ma.  The current draw varies with temperature, voltage, and part to part, thus the datasheet specifies a range.

The current specifications are in the data sheet so that the designer can add up all of the currents for all of the parts in the design to determine the output current required from the power supply.  In your design, as long as the 5V power source can provide 64ma, you shouldn't have a problem.

Because of the resistor in your circuit, the voltage at the Vcc input will not be 5V due to the drop caused by the current drawn by the part.  You need to remove the resistor before the circuit can work.