Real Vikings never wore horned helmets. They did wear furs when they were outdoors in the winter, but inside and during the relatively warm summer months the medieval Norse people wore tunics and dresses very similar to the rest of the population of Europe. The fabric may have been a little thicker, and the embroidered trim may have looked a little different, but the filthy barbarians depicted in certain--albeit hilarious--"What's in your wallet?" commercials are a product of Hollywood.
Choose your fabric. Look for natural fibres--wool or linen (or very good faux wool or linen). Cotton and cotton/poly blends should be avoided if you're trying to be authentic. For colours, stick with browns, greens, reds or blues for overtunics and breeches; white, off-white and ivory for women's underdresses.
Make the men's clothes. For everyday wear, Viking men wore a pair of trousers that tied at the waist and fit snugly against the leg. A modified pajama-bottom pattern would be a big help in making these.
The trousers were covered with a knee-length overtunic or T-tunic. Measure your man from the shoulders to the knee, and double the measurement. Buy that much 44/45-inch-wide fabric and fold in half (matching the selvedge edges); the fold will cover the shoulder on the finished garment . Measure a foot down from the shoulder fold) all the way across the fabric, and mark with pencil or chalk. This denotes the sleeves.
Measure the chest, add about four inches, and divide in half. Mark this half measurement on the sleeve line, centring the measurement on the fabric. This is the chest of the tunic. From each end of the chest measurement, draw a diagonal line to the bottom corner of the fabric. You should have a "T" that flares at the bottom drawn on the fabric.
Sew the sides and sleeves, and hem the sleeve ends and tunic bottom. Cut a hole for the head and hem. Add embroidery or trim if desired, as per the links below.
Make the women's clothes. For the undertunic, measure from the shoulder to the floor and double. Make a T-tunic as above with the white or ivory fabric, but use 60-inch-wide fabric so the undertunic has long sleeves.
Add a colourful overtunic, using the same instructions as above, but make the tunic mid-calf length, rather than knee length.
Dress the kids. Norse children basically wore smaller versions of adult clothing, so adjust the above measurements to fit the kids. Very small children can be dressed in just overtunics for convenience in diapering.
Always wash and dry your fabric before you begin to cut and sew it.