Stream of consciousness
This exercise is for developing the ability to be aware of what's going on in you. This can be quite helpful with working through your feelings.
For this exercise, you describe everything that comes up in you. When you notice a particular feeling or thought, you describe it, and then you return to feeling and noticing yourself.
Try doing this without judging what you feel or think. There is no right or wrong here, there are feelings that come up. When you find yourself judging anyway, note that, but don't take it seriously while doing this exercise.
You can describe things by writing them down, but you can also speak them. There are a few ways that you can do this, just see with which you feel the most comfortable.
Note that there is no need to keep your descriptions. The point of the exercise is to be aware of the feelings and thoughts that go on in you, without acting them out. You feel them, rather than do them. It actually helps the working through process to not go back to your descriptions. So, you can delete or throw away writings or recordings afterwards.
Writing things down is convenient, as you can do that on your own. You can use a pen and notebook, or a computer, a tablet or phone, anything that you can write with.
A disadvantage to writing things down is that it takes some time to do, and can thus distract from your feelings. It would therefore be a good idea to choose a way of writing that goes quickly and easily for you.
Describing your feelings by speaking is easier and faster than writing things down. You can still do that on your own, but if you need the feeling of someone listening, you could record what you say, or have someone witnessing.
Another advantage is that you hear your own voice, and the feelings that you express in your voice. That offers another way of becoming aware of your feelings. You can also do this with your eyes closed, which for some helps to be more aware of their inside. It is, however, a good idea to experiment with closing or opening your eyes. You might want to check whether you can still feel your facial expressions as good as you need to, with your eyes closed.
The disadvantage to speaking aloud may be that others can hear you. Since this is basically about the most intimate of your feelings, that is probably not what you want.
Having a Witness
It is possible to do this exercise with someone else, who is called the "Witness" or "Listener." Some people feel more at ease describing their feelings to another person, rather than writing them down, or speaking to noone but themselves, so that could be a reason to do this.
If you have someone there who listens, they should be instructed to listen only. They are a witness to what you tell about your feelings, not a participant, and should not interfere. They should especially not judge you or your feelings, or at least not show that. It is important that you feel as free as possible to feel whatever comes up in you. Even if you feel like killing someone, it is fine to feel that, as long as you don't act out those feelings.
Only if you do this with a therapist or spiritual teacher, can they interfere, if you want to allow them to. This is because the exercise is for you to develop more awareness to your mental processes as they happen. When anyone interferes, the process not only changes, as it will also be about your feelings about the other person, but you may also get to feel less free to feel and describe your feelings.
You can actually work through feelings as they come up, without describing your feelings. You can just feel them, and take note of them in the same manner as if you were going to describe them. The important point is to not act them out, to only feel.
Especially in the beginning, however, describing your feelings may be quite helpful to avoid getting distracted. You may more easily get lost in daydreaming or worrying, if you don't do that. It is therefore recommended to start out with describing, and later move on to doing the exercise by only feeling.