There are a lot of languages. I use Python. Right now I'm working on a project using the Pennylane library for quantum machine learning. Other languages exist – dedicated ones like Q# or quipper (built on top of Haskell) or libraries in Julia, C++, etc.
One paradigm, variational quantum algorithms (part of the broader class of hybrid quantum-classical algorithms), has several similarities to classical deep learning. The approach can be summed up as:
1. Construct a parameterized quantum circuit – one where the quantum gates are controlled by real-valued parameters. Ideally this circuit is one you think could reasonably solve some problem based off of your knowledge of the problem, input data, and behavior of quantum information.
2. Define an objective function.
3. Iteratively adjust the circuit parameters. This is done by measuring the output of the circuit and using that information in conjunction with some optimization function (e.g. stochastic gradient descent).
In general terms, I'm working on QML algorithms that may some day be used for applications in biology and medicine.