Find an example of Old English text. While Old English printed texts are available (certain copies of the Old English poem Beowulf are printed in the original text), you can just as easily find copies of the alphabet online such as at the website Ominiglot (see References). Print out a copy of the alphabet to act as a reference.
Place a sheet of tracing paper over the reference sheet. Most of the upper case Old English letters are nearly identical to modern English ones; however, several of the lower case ones have their own distinct look. The letter "s" has three lower case variations and the letters "g" and "w" barely resemble their modern versions. Tracing the letters from the reference will help you train yourself to form the letters properly.
Practice writing the letters free hand. Once you feel comfortable tracing the letters, write the letters on your own. The upper case letters will likely be the first you master since they're nearly identical to the modern script. The challenge with some of the lower case letters will be remembering to orient them properly. When writing modern English, the lower case "f" stands parallel to its upper case version. The Old English lower case "f" drops down lower like the lower case "p." While it might not seem significant, placement and orientation of the letters is part of what separates Old English script from modern English.
Replace your regular pen with a calligraphy pen and practice writing the alphabet this way. Pen nibs from the 8th century created lines similar to the way that calligraphy pens do. While you can write perfectly legible Old English script using a ballpoint pen, use a calligraphy pen to make the script appear more authentic.
Write out entire words in Old English. The website Omniglot (see References) has examples of Old English text written in both the original script and a slightly modernized version of Old English. To make it a bit easier on yourself, start with the modernized Old English. Once you feel comfortable writing in that style switch to the original text form.