Yes, you can play as a single player!
However, Helldivers does not have a story mode or single player campaign, as it was designed to be played with friends.
Difficulties 1 - 5 can reasonably be completed solo, but planets above that difficulty can be a real challenge for a lone Helldiver - but not impossible.
You can find community written guides and tips for solo Divers on the forums. For example, here's a link to Greywolfs' Solo Diving Guide.