According to Rabbi Moshe Chaim Luzzatto, in his work Mesilat Yesharim (1:26) the primary purpose of man is to:
fulfill the commandments, serve [G-d] and stand up to trials.
In 1952, people asked the Lubavitcher Rebbe "What is the purpose (תכלית) of life?". The Rebbe answered them:
To bring light (ליכטיקייט) into the world.
Similary, the Rebbe states:
When one has ליכטיקייט then he will have found the תכלית. A person cannot feel that which he lacks. One looks for what he does not possess. If he had what he was lacking for he would not search.
On 20 Adar 1, in the year 5711 (26th February 1951), the Rebbe writes in one of his many letters to a person, what exactly is our purpose in life. The first lines are the most important to this specific question:
The life’s purpose of every Jew, man or woman, has been clearly defined as far back as the Revelation at Mount Sinai more than 32 and a half centuries ago, when we received the Divine Torah and became a nation. We were than ordained as “a kingdom of priests and a holy nation.” This means that every one of us must be holy in our private life, and in our association with the outside world every one of us, man or woman, must fulfill priestly functions. The priest’s function is to “bring” G‑d to the people, and to elevate the people to be nearer to G‑d. Similarly, every Jew and Jewess fulfills their personal and “priestly” duties by living a life according to the Torah.
The extent of one’s duty is in direct proportion to one’s station in
life. It is all the greater in the case of an individual who occupies
a position of some prominence, which gives him, or her, an opportunity
to exercise influence over others, especially over youths. Such
persons must fully appreciate the privilege and responsibility which
Divine Providence vested in them to spread the light of the Torah and
to fight darkness wherever and in whatever form it may rear its head.
This is your duty and privilege as one of the student officers in
relation to your coreligionist colleagues and student body in general.
I would also like to convey this message to your colleagues in the
JCF. You are all no doubt aware of this, but perhaps there is room for
added emphasis and the conviction that “it cannot be otherwise.”
See also: Likkutei Sichos, Volume 15, Bereishis, Sicha 3 (a text-based translation and study can be found here).
But, back to your question: what are we supposed to achieve in our lifes? The Rebbe explains that we are supposed to be "priests". "We must fulfill priestly functions. The priest’s function is to “bring” G‑d to the people, and to elevate the people to be nearer to G‑d. Similarly, every Jew and Jewess fulfills their personal and “priestly” duties by living a life according to the Torah.". That is our purpose in life, and that we are supposed to achieve. Also to be a light unto the nations and spreading G-ds holy word, His Torah.
And through Torah and Mitzvot the person transforms his desire to receive into a desire to give, and becomes a "receiver in order to give". That person can now receive all the goodness from G-d, and at the same time have a strong connection with Him, because through Torah and Mitzvot he has acquired similarity of form to his Creator. This stage is called " the last state of tikun", or " the thought of creation". Now there is no longer a need for the impure "other side". When this occurs, "death will be no more". Being diligent in Torah and Mitzvot during these "6000 years" and in a person's lifetime, is for them to attain this final stage.