Well, technically the language of the Vikings was Old Norse that evolved into two similar dialects: Old West Norse (Norway and Iceland) and Old East Norse (Sweden and Denmark). So we can see Swedish as a lineal descendant of the Vikings' language evolved some 10 centuries later.
The excitement you feel when imagining how cool it is to speak/learn the language of these mighty seafarers (though it is a stereotypic propagated image) doesn't end there. It is history now and has little to do with modern Swedish alphabet, which adopted Latin alphabet in 11th century, but before that the dialect of Old East Norse people spoken in Sweden was called Runic Swedish. And it was "runic" because the body of text appears in runes!
The Scandinavian alphabet was also known as futhark or fuþark (derived from the first six letters of the alphabet: F, U, Þ, A, R, and K). Old Norse was written with the Younger Futhark alphabet, which had only 16 runes (unlike Elder Futhark with 24 runes).
Todays' "Vikings" have their alphabet being identical to the Finnish alphabet. It consists of 29 letters: 26 letters are the same as in English, plus Å/å, Ä/ä, and Ö/ö at the end.
No wonder that for the Swedes learning English is easier than, say, for the Germans or the French: there are quite a lot of similarities between these two languages as they share many common historical linguistic influences and are shaped by Old Norse, Latin and other Germanic languages. They are alike in grammar, and though Swedish definitely wins by the low amount of tenses (6 versus 16), it has cases for nouns and more complicated rules for articles.
Swedish is not a widely spread language. It is spoken by about 10 million people, 90% of which live in Sweden. It is very close to the Danish language, but there is an inherited cultural competition between Sweden and Denmark, countries with a long history of wars, so over centuries various changes were intergrated in both languages in order to differ.
Yeah, just blame the Danes and speak Svenska, which dawn the Vikings marked in runes!
Kram! (Hug!)
