Vikings are a perennial favourite among children's costumes. They are highly recognisable and easy to put on and wear for long periods of time. There are many commercially made Viking children's costumes available, but the costume is not difficult to make at home, which can allow you to more fully tailor the costume to your child's individual fit and tastes.
Start off you child's costume with the clothing that will form its base. Find either a long-sleeved shirt or a shirt you can remove the sleeves from entirely and loosefitting trousers. The shirt should be significantly larger than your child's shirt size, so that it comes to mid-thigh to resemble a tunic. Both of these items should be solid, muted colours; browns, tans and greys work best, but other colours will work as well. Avoid clothes with obvious modern features, such as buttons or stretchy collars, or remove them from the clothes. If your child doesn't have clothes that can work for this, suitable items are fairly easy to come by at thrift stores.
Try to find plain leather boots or shoes for the child to wear. If you cannot find any, most nondescript brown shoes will work in a pinch. If you do find boots, tuck the trousers into them rather than letting them hang down over the boots.
Find a child's vest at a thrift store and sew faux fur, available at craft stores, around the edges or even over the entire vest. If the vest looks particularly modern and you do not want to cover it with fur, you can also stitch a layer of leather or plain cloth over it.
Find a plain leather belt, and have your child wear it over the shirt at the waistline.
The most easily recognisable feature of a Viking costume is the horned helmet. Pre-made helmets are widely available at costume stores, but you can also make your own. Use the bottom half of an empty milk jug for the base of the helmet, then make horns by crumpling newspaper into the right shape and wrapping it completely with masking tape. Tape or glue the horns onto the sides of the helmet and cover the rest of it with aluminium foil.
No Viking costume is complete without a weapon, usually a sword or axe. Again, these are easy to find at costume shops, but they are also easy to make by gluing together several layers of paperboard cut into the right shape, then painting the "blade" grey or silver.