Trees take up water from the soil and cast shade. No plants will thrive in soil that is always dry and shady, except perhaps the common ivy, but if the ground gets at least some sunlight, there are things you could do.
1. Choose the right plants
Like those that naturally grow in woodlands. They are tough enough to put up with the competition. There is aucuba, mahonia, euphorbia, epimedium and many others, even a few ferns and grasses. You will find a very useful list here www.rhs.org.uk
2. Plant in the autumn
This gives the new arrivals time to settle in before the trees spring back into growth. Use plenty of garden compost or any soil improver from the garden centre, mix it with the soil before you return the mixture to the planting hole, water well.
3. Spring-flowering bulbs
When planted underneath deciduous trees, they will put on a great show before the leaves arrive on the trees above. As well as snowdrops, daffodils, and English bluebells, there is cyclamen and dog’s-tooth violet (Erythronium).
4. Mulch between plants
Spread a thick layer of garden compost or chipped bark on the soil between plants, and be prepared to water in dry weather, at least in the first year.
Epimedium and native bluebells do well under trees