“Life is suffering” is a misleading way of putting it. That makes it sound like we never experience joy or pleasure, which, for most of us at least, is manifestly not the case. The common experience is that life is a mixed bag of ups and downs.
A better way of stating what the Buddha was getting at is that nothing is capable of giving lasting satisfaction. His original teaching was that all phenomena are marked by “dukkha”, a difficult-to-translate Pali word that can mean simply “unsatisfactory” or “stress-causing”.
This is closely linked to his other teaching, that all things are impermanent; things come and go. This constant shifting means that we can never find lasting ease in any particular set of circumstances. Even when things are “going great”, we’re aware of the precariousness of the joy that comes from those favorable circumstances, and that it can’t last. That causes unease—dukkha. And of course, when things are going badly, we can’t wait for things to change.
The gist of the teaching, then, is that we find ourselves constantly wishing either that pleasant things will stay the same (which they can’t), or that unpleasant things will go away. This wishing is the “desire” part of your question. We’re always wishing things were otherwise, fighting the natural way of things. Thus our inability to find lasting ease.