Try this:
print(j, end='')
end by default is \n (see print()). Also, be sure to print a newline at the end of each outer loop iteration:
for i in range(1,6): # notice that I changed this to 6
for j in range(1,i):
print(j, end='') # added end=''
print() # printing newline here
1
12
123
1234
EDIT I just noticed you were using Python 2.7. Since that's the case, you can use print j, instead of print(j, end='') and print instead of print(). Note that print j, will leave spaces between the js. If you don't want this, you can import sys and use sys.stdout.write(j) instead (see sys).
Furthermore, if you want to use the Python 3 print function as shown above, you can always
from __future__ import print_function