Key points to know before you plan -
1. People diagonsed with Asthama ( or any kind of Respiratory disease) and Heart disease should refrain from visiting this lake as it can be a pain for you - I have seen people struggling to breathe and fainting throughout my drive to the lake. Even after reaching the lake, many faced problem.
2. If you are one of those unlucky people, whose stomach isn't a good partner when driving on mountain roads (you tend to vomit a lot) then don't choose to go for a shared tour because they take you in TATA Sumo or Mahendra Bolero (10 tourist per vehicle) and drive quite rough. Plus eat very light as long as you are on this route.
3. Shared tour to North Sikkim (Lachen, Lachung - including Gurudongmar Lake, Yumthang Valley , Zero Point) would cost around Rs.3600/- including fooding and lodging. Private tour cost starts from Rs.18000/- per vehicle.
4. Many tour operators give tourist an option to visit Kala Pathar at an addition cost or in place of Zero Point - but my personal suggestion is if you have to choose between these two , go for Zero Point. Kala Pathar is just some 2-3 mountains covered with snow. However you can skip this if playing with snow isn't your thing.
To give you context, let me share a pic of Kala Pathar -