There is parking for about 6 cars at Lough Annascaul. Start by following the roadway up the valley until it peters out in the bog above. From here head NE initially across bog and then continuing NE gradually ascend the southern slopes of Beenoskee, but actually heading for the neighbouring Stradbally Mountain. You will eventually reach a fence which will lead you up to the summit of Stradbally Mountain and its impressive views. From here it is a 10 minute walk west to the top of Beenoskee. Beware there are major cliffs on the northern side of this ridge. From Beenoskee you can descend SW and pick up the track back down to Lough Annascaul. Allow about 5 hours.
An alternative route starts from Glanteenassig Lake. This is well signposted off the Tralee - Castlegregory road. Follow the boardwalk along the western side of the lake to where a stream/waterfall descends into the lake. Follow this stream upwards onto the plateau above. Turn right and it's a steady climb up to the summit. I would avoid this route in icy weather. Linkback: https://mountainviews.ie/summit/42/comment/4802/