Hmmm... the "core" (ahem) of SMPS design is the transformer (the magnetic components in general) so saying that's "all" you need help with is basically saying "design it for me".
That said, the phasing dots traditionally indicate the start of a winding and in this case should be assigned to pins 3, 5 and 10 of your schematic (alternatively, pins 1, 4 and 7 could be the dot ends, but this puts the swinging node of all the windings closest to the outside which increases noise emissions).
The EFD25 core size is a bit big for a 15W/>100kHz power supply but its always possible to use a bigger core, whereas the converse is not true. Regardless of which size you choose I would strongly suggest getting a ferrite core set from one of the major electronics parts distributors (e.g. - DigiKey) which has been pre-gapped to a specific A
L value (which gives the inductance in nH per turn squared; e.g. - a 100 turn winding on a core with an A
L of 250 results in 2.5mH inductance). Note that the A
L value for a pre-gapped set assumes the use of one ungapped and one gapped core half. For this frequency range EPCOS/TDK N87 material is ideal and is widely available.
The TNY280 runs at 132kHz and has a 700V switch so for a max 265VAC input a Np:Ns turns ratio in the range of 30:1 to 40:1 would be good for a 5V output - turns ratio is not terribly critical in a flyback, btw. The primary inductance should be somewhere between 2mH to 2.5mH.
As per usual with SMPS noobs, I highly recommend tinkering around with the calculator for a flyback converter on this site:
http://schmidt-walter-schaltnetzteile.de/smps_e/spw_smps_e.html.
At that site, plug in 280V and 360V for the min and max Vin, respectively, and manually set L1 to 2.4E-3 (2.4mH) and N1/N2 (aka "Np:Ns") to 30. Click on calculate, then on transformer data and scroll through the selections. Note that you can add cores to the table manually and it will remember them for your browser (using cookies, I guess?) for quite some time. For example, plugging in the data for an EFD25 core with an A
L value of 250 (one of the stock gap values available) results in a primary with 98 turns and a secondary with 4 turns.
Don't forget you need 3 layers of mylar or kapton tape between primary and secondary. I would use a higher clamping voltage for DZ1 - it needs to be higher than Vo * Ns:Np by a good margin for the flyback converter to even work - or else skip it completely and rely on the RCD clamp alone.
On a related note, you want C10 to bridge pins 1 and 7, rather than 1 and 10, and note that this is a safety-critical component (needs to be "Y2" rated).
Your feedback network is messed up, but I'm done for now.
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