Almost any normal sorbet recipe will contain a decent amount of sugar, and strawberries are no exception. I'd guess probably 1/2-2/3 cup per pound of strawberries. Use a substitute if you have an aversion - honey, agave, raw cane sugar, whatever you prefer. (Of course, anything liquid is going to contain some water, and cause a bit of ice, but it's still doable.) A sorbet without any extra sugar, even when frozen normally in an ice cream maker, is going to end up with a very icy texture, which will probably keep you from experiencing the flavor as much. The sugar helps soften it.
The other common way to soften a sorbet is alcohol. Rosé wine (thank you, David Lebovitz) works quite well in a strawberry sorbet. Vodka can be your go-to liquor for any impromptu sorbet with other fruits, since it'll add alcohol without any flavors that clash. Beyond that, either look for recipes, or look for mixed drinks containing the fruit you're using.
You might also be having problems simply because your strawberries aren't that great. Did you freeze them, or were they storebought? If you bought the strawberries yourself, you'd know they were decent before freezing, while not all storebought frozen fruit is as flavorful.
Finally, it's possible that your frozen strawberries collected some extra water in the form of frost, which then makes your sorbet icier. If there are obvious big chunks you could scrape them off.
The problem could be the recipe. If you're using a lot of lemon juice, then a lot of sugar to cover up all that acidity, you're guaranteed to end up with something pretty soft. Have a look around at other lemon sorbet recipes, and see if you're using relatively more.
But if your recipe is sane, then it could be your process. Here's how it could mess things up.
Sorbet is basically sugar water with a bit of flavor. When it freezes all the way, unless it was churned as it froze, there's going to be a tendency for it to separate into ice crystals and even stronger sugar water. Some of this may happen in your first freezing step. Then you're pulverizing it in a food processor, probably giving it a chance to melt a bit. This is much worse than just leaving it out, since once it's in tiny bits, there's a lot more surface area exposed to air, and you're continuously mixing it, bringing in warm air, and so on. Then finally, it refreezes. The part that managed to melt in the meantime is going to repeat the earlier process, except now there's lots of space, so it's even easier for ice crystals to form and force out the sugar. So I'd definitely expect a tendency to end up with a runny sugar syrup component to your "sorbet". (If it's at all grainy, this is probably what's happening.)
If you're determined to do this without any kind of ice cream maker, it's going to be hard to avoid problems. You can certainly try to correct for it by using less sugar, though. You could also just make a granita - freeze it on a sheet, and scrape it off to serve.
Best Answer
Cold temperatures make you perceive flavors less in general. You can notice this quite directly when well-frozen eating sorbet or ice cream: you'll taste the flavor much more strongly initially than later on. And it's not just you getting used to the flavor. If you switch to something else, you'll still have a dulled perception of its flavor. I believe this is all primarily because your tongue and palate are numbed a bit. Your sense of touch doesn't work as well if your hands are freezing either! The lower temperature also reduces volatility, so aromatic things won't be as readily available to perceive.
As for sweetness, there's not exactly a rule of thumb per se for sorbet. I guess it's common to be somewhere in the ballpark of 1/2 to 2/3 a cup of sugar for 2-3 cups of fruit, but that's often more about adding enough sugar to keep it soft than anything else. In many cases, you end up adding lemon juice or something else to cut the sweetness back down, or perhaps using less sugar but adding alcohol to soften it.
In any case, you can certainly develop a sense of how sweet your sorbet will be before freezing, but it's hard to really convey in words. It's just a lot sweeter than you want it to taste in the end. Tasting a known good recipe might help train you more quickly, or perhaps tasting melted sorbet - but in that case, if you've just eaten a bunch of it, your perception is probably a bit skewed.