Hmmmm...well, the normal way of improving clay soil is to dig in lots of horticultural grit and organic matter, then every time you plant something, dig the hole larger and backfill with compost....however, if it's so badly drained, I think you'll have to consider installing drainage pipes

The only other solution would be to create a bog garden, and plant things that enjoy having their feet wet, such as Gunnera, Rodgersia, Ligularia, some Primulas (candleabra and florindae), Lobelia cardinalis, Lythrum salicaria, Iris laevigata and ensata, Calla palustris, Zantedeschia aethiopica and so on. There are lots!