The Hebrew word for righteousness is tzedek. Tzedek is almost impossible to translate, because of its many shadings of meaning: justice, charity, righteousness, integrity, equity, fairness and innocence.
Because of this, if we are to have any hope of how to exemplify righteousness in the world in the way that Hashem would desire, then we must have the proper lenses by which to view how we are to act in many situations so as to allow righteousness, or tzedek, to shine. This meditation below is a common practice in many communities around the world.
(Note: For more information on meditative techniques, see the source Jewish Meditation by Aryeh Kaplan.)
Find a comfortable seat in a quiet place. First, take a few deep breaths. Deep breathing brings oxygen to your brain and helps clear the mind. Close your eyes. Continue breathing until you feel relaxed.
The focus of this meditation will be upon Hashem solely. Using a well-known phrase for meditation in Judaism, choose whether you will say the Hebrew or the English. Both are provided below:
:שִׁוִּיתִי יְהוָה לְנֶגְדִּי תָמִיד
(Shiviti Adonai L'negdi Tamid)
I have set Hashem before me always. --Psalms 16:8
The image included in this article is an example of Shiviti, which are meditative plaques used in some communities for contemplation over Hashem's name. The Name, or Tetragrammaton, is never said - however, this part of the psalm above as transliterated is used. Quietly speaking this phrase, or meditating upon it without words in your mind, picture the image from this article and completely let the concept of Hashem, His Name, and His being before you consume your thoughts. As you repeat this phrase over and over, you will find that the idea will change, be absorbed and affect you in a completely new way. Don't try to control it - let Hashem work in you. Don't try and meditate on the concept of righteousness - the repetition of this verse is the focus.
True righteousness will only come from having Hashem before you at all times - like lenses, to help you to see the world, your actions and your duties in the proper way.
Meditate on this phrase for at least 10 minutes without interruption. As you slowly and gently end your meditation by opening your eyes, try to maintain a sense of quietness for a time after - allow the experience to flow through you and feel its effect. By meditating on the Divine, you will feel less inclined to pursue your idea of righteousness, but rather that of the Eternal.