By far the easiest way to Nyika is by air. Air Malawi used to run scheduled flights to the Chilinda airstrip via Mzuzu, but now they've stopped and the Nyika Safari Company offers flights between Mzuzu and Chelinda for US$240 one-way for the plane (takes up to 5 people). The views from the aircraft are stunning, if you've picked a clear day with few clouds, and you can clearly see the foothills, the patches of natural forest, the plantations in Linnaean symmetry and the open expanses of rolling heathland.
Coming from Malawi, Chilinda Camp lies roughly 120km from Rumphi, and is reached along roads which shouldn't present any problem to a 2WD in the dry season but which will require a 4WD vehicle after heavy rains. The route is clearly signposted: from Rumphi you need to follow the S85 westwards for roughly 50km, then turn right into the S10 to Chitipa. Thazima entrance gate is 8km along the S10. About 45km past the entrance gate, a signposted left turn-off leads to the Zambian Resthouse, which lies about 2km from the S10. Perhaps 50m further towards Chitipa, you'll see the turn-off to the right signposted for Chilinda Camp. Chilinda lies about 16km from this turn-off.
The best place to stock up on food before you reach Nyika is Mzuzu, and the last place where you can be sure of fuel is Rumphi (although you can often buy fuel at Chelinda during the dry season). The drive between Lilongwe and Nyika cannot be done in a day during the rains, and it's a very long (9 hours) slog even in the dry season.
There is no public transport all the way through to Chilinda. In the dry season, you can try hitchhiking but the best way is to catch a matola from Rumphi that is going through the park to Nthalire or Chitipa. For a small fee they will divert to Chilinda; or for approximately US$70, you can hire a minibus all the way.
If you want to hitch, you'd be wiser trying to do so from Rumphi all the way to Chilinda, and be sure of not getting stuck halfway. The best days to hitch are Fridays and Saturdays, which is when Malawian residents tend to head to the park (but also when accommodation is most likely to be fully booked – not a problem, of course, if you have a tent).
Another option is to ask the National Parks and Wildlife Office Service in Mzuzu whether they have any vehicles heading to Chilinda, bearing in mind that such vehicles are infrequent and very often too loaded with park staff for tourists to be offered a lift.
One of the options for backpackers leaving the park is to hike to Livingstonia over two or three days.