Look closely in the fruit and see if you can find a tiny pink worm, or maybe a white one. They are either Plum moth or Plum sawfly.
The black on the leaves sounded like one of the common fungal infections, spray with a suitable fungicide at the time advised on the container.
Picture of the grey mould would be useful, but it could just be lichen which is harmless.

Any chance of photographs - of the mouldy tips, the infection on the leaves, and the fruits? I know you can't attach them to your question, but if you put them in your Photos and then add a note to this thread, we can have a look at them.
Do the leaves which aren't blackened have patchy yellowing or are they ordinary, healthy green?
Is there any blossom which died back and remained on the tree afterwards?

Lots of advice on plum tree problems on RHS website (amongst others) http://www.rhs.org.uk/Gardening/Grow-Your-Own/Fruit-A-to-Z/Plums,-damsons---gages

Everything from silver leaf, canker, plum moths, aphids... the list goes on and everything is pretty gross that it makes me think growing your own plums is far more trouble than they're worth. I work with someone who grows them, brings in loads every year and probably 3/4 get thrown out because no-one eats more than one or two and they rot in the box/bowl.

Basically, prune at the right time, cut out infected wood/ bark/ leaves/ fruit and burn, set up pheromone traps for the moths / wasp traps for the wasps / nets for the birds and on and on.

And all has to be done at the exact right time and not a day either side!