I've carried stuff in all sorts of manner, from double-bagged plastic grocery bags on the handlebars, to the backpack fully loaded to panniers for my now-regular 30-mile round-trip grocery run. I've even stuffed stuff into the back pocket of a jersey (most recently, a half-litre carton of milk on Saturday), or even under a shirt tucked into my trousers.
If you haven't got panniers, the backpack is the obvious way to go. The one I have can be stuffed into its own pocket and strapped under the seat if need.
Next time, HLM, what about a pillowcase?
People are afraid they are going to damage food when transporting it on a bike. The classic concern, of course, is eggs. I have found them to be a lot tougher to break than at first thought. The last dozen lot came home over 25km this past weekend in a bag on top the rear rack that also contained three loaves of bread. Not a problem! Even with 25C tyres at 80psi and some roughish bitumen.
I've also found that unless the temps get up to 30-plus deg C, chillled and frozen goods will do quite nicely packed together in a pannier and travelling for around 90 minutes before being tossed into the fridge at home.
So fear not anymore. Grocery shopping really is a cinch by bicycle