This is a guess......

Im not sure if Blackberries need certain chill hours to set fruit. I do know raspberries can have problems if the winters arent cold enough. I think most of greece is zone 8 or 9. You may just have not gotten the right type for your climate?

Also, yellowing leaves is ususally an indicator that something is missing in your soil. That could be iron, or nitrogen. Also, I do think Blackberries need slightly/acidic soil, which can cause chlorsis (yellow leaves).

I think a soil test is in order.

You have classical symptoms of sunlight UV damage to the fruit (the fruit is setting properly, it is just not ripening properly). Greece is a very sunny place, at least it looks that way in all the movies ever shot there.
It is not a systemic disease acquired during shipping.

Move your plants to a more shaded area if possible or try using some shade cloth (next year). Reflected light from walls would be bad also (lots of white walls in Greek movies).

If the red druplets on the otherwise ripe fruit are very hard, research "red berry mite".

Your leaves with the burnt edges and the late primocane sprouting may be symptoms of not enough water. Perhaps you water a lot, but the soil will not hold moisture. Perhaps it is hot all the time. Try putting a few inches of compost atop the soil to hold water better.

It is not a chill hours issue. The advised soil test is not a bad idea.

thanks for the replies. I'll see what I can do about moving it to some shade to see if it helps with the rest of the berry production as it's only just started to produce. I'll also look in to a soil ph test. I didn't realize that blackberries liked acidic soil like blueberries. I have some blueberries that are doing well, first year for fruit from them as well, but every once in a while I add a little vinegar to the pitcher when I water them. Maybe I should do the same thing for the blackberries? I have also used an all natural fertilizer made from bat guano so I don't think nitrogen is a problem here though I did only recently start using that as of 3 weeks ago.

Don't use vinegar solutions on the blackberry and don't go overboard with the guano, a little is fine. Blueberries prefer a ph of around 5 while blackberries do fine in neutral soil.
Don't move your blackberry plant during fruiting and active growth, that is for autumn/winter when growth has stopped. Shadecloth or any sun-blocking method is fine. Do not let dark cloth touch the berries, they will cook.

You actually have nice-looking fruit except for the sun problem. It is ironic that a plant that otherwise wants sun should have such wimpy fruit, but that is the case with certain newer blackberry hybrids. My berries get the same problem, but not as bad as yours.

OK, glad I didn't jump the gun and add vinegar to the watering last night. If you look at the other pictures in the link I provided in my first post
http://sdrv.ms/Mr2uR7
you'll see that the blackberry bush is in a container on my balcony so moving it won't be a problem ;) I placed it in the corner of our patio that gets the most sun because I thought the sweetness of the berries was directly related to the amount of sun it gets but I guess the temperatures here in Greece are just too hot for it to be in full sun most of the day.

About the type of the blackberry bush. It's not a new hybrid, I can't say exactly what it is, all I know is that it's a start from a bush that was planted in my grandparents backyard at least 50, maybe 60 years ago, maybe longer. Since then, every time we moved my parents always planted a start from it and now I have it here with me in Greece.

I did have another question about fertilizer. I think I've gone easy on the guano. I've only added it once and I don't think I added too much. Since you said to go easy on the Guano, are there some side effects of using too much of it? What fertilizer would you recommend? I know that because it is in a container that it should need to be fertilized a little more often than a regular plant since the nutrients basically get washed out the bottom when it's watered. I'd also like to mention that when I planted this in the container I added some of those water crystals to the soil to help in not only retaining the water but in the hopes that it would also help retain some of the nutrients as well.

Here is a link that might be useful: more pictures of my blackberry bush

You need add liquid iron use feed grade liquid iron spraying plant want help this year buy soil application will help next year growth to have iron needed pot and newly planted blackberries need dose iron in small amount one teaspoon liquid iron in gallon sprayer with water will correct low iron it takes iron to make black color in berries and green in leafs one plant needs teaspoon iron your plant need heavy dose but once every 3 years plants that are in ground. If you do this all drulets be black next year it will amaze you the turn around.

I didn't see a pot in any of those pictures.

A blackberry in a pot in Greece, now that explains the browned leaves!! You will have to consider your plant an experiment. The roots can get as warm as the surrounding air unless you keep the soil continuously damp, quite a chore. Try some of Gator's methods, he knows plenty. You've already done pretty well with the plant, considering the circumstances.

You were originally wondering about growing from seed. Any resulting plant will probably not have identical fruit. The fruit may be better or worse. "Starts" are better.

@ Gator - Thank you for the information regarding the Iron. That would explain a lot, I really hope it works. With my blackberry bush being in a pot instead of being planted in the ground, how often would you recommend that I add liquid iron? From what I've read plants planted in pots need a little more TLC since the nutrients can get washed out when watered.

Also, what about the fact that some of the druplets look like they are "bleeding" and later drying out? Is that due to the lack of iron as well or are they just very sensitive to the heat we have here in Greece? As a precaution I have moved the blackberry bush to the opposite corner of our patio which still receives full sun but not for as many hours as the corner of the patio it was in.

@ Larry - The folder with the pictures of it in a pot is easy to miss due to the way Skydrive's default layout is. The top left "icon" is a folder labeled "first years growth" which is rotating through the pictures inside it so it can look like just another picture.

Do you guys maybe have a suggestion outside of keeping the soil damp that could help me keep it cool? Like if I was to create some kind of a box around the outside of the pot to keep the sun from directly hitting it and burning the roots? Maybe not closed off completely but something that would provide shade around the pot but still allow air flow so as not to create an "oven" effect potentially keeping the pot cooler.

Again thank you for the help and suggestions!

The bleeding is part of the UV damage process.

OK, I found the pot. Your setup is ideal if you can keep the pot located where it is in the shade of the low wall or railing. Otherwise, just prop a reflective chunk of something on the sunward side of the pot to shield the whole pot.

You may have too much berry and root system for a pot that size. Of course a larger pot becomes harder to move around, but there are simple roller devices for moving large pots.

Once your plant has set little green berries, it can be moved into complete shade. The fruit will ripen anyway.

Don't rely on just pouring water on the plant as a gauge of how wet the soil is. Probe deeply with your fingers on occasion or get some kind of moisture sensor to probe in the soil. Often potted soil is very unevenly wetted to a significant depth, even when watered evenly from above. Water tends to go to the inner pot surface, deflected by upper roots, leaving lower roots dry.