Serial sends data 8 bits at a time, that translates to 1 byte and 1 byte means 1 character.
You need to implement your own method that can read characters into a buffer until some sentinel is reached. The convention is to send a message like 12431\n indicating one line.
So what you need to do is to implement a buffer that will store X number of characters and as soon as you reach that \n, perform your operation on the line and proceed to read the next line into the buffer.
Note you will have to take care of buffer overflow cases i.e. when a line is received that is longer than your buffer etc...
EDIT
import serial
ser = serial.Serial(
port='COM5',\
baudrate=9600,\
parity=serial.PARITY_NONE,\
stopbits=serial.STOPBITS_ONE,\
bytesize=serial.EIGHTBITS,\
timeout=0)
print("connected to: " + ser.portstr)
#this will store the line
line = []
while True:
for c in ser.read():
line.append(c)
if c == '\n':
print("Line: " + ''.join(line))
line = []
break
ser.close()