Located in the present day Parco archeologico del Colosseo, Palantino (Palantine Hill) is just up…read morethe steps on the partially man-made slope with origins from prior to common era.
While the hill is formed of natural sediment, its sides, faces and top have been significantly altered by human activities, most notably by order of a succession of emperors including Domitian and Augustus. The imperial palace(s) built into the rock fell under disrepair several hundred years after their initial construction, later to be built over in the Renaissance by wealthy families like the Farnese who embellished the hill with artificial terraces, gardens, and grottos.
Looking towards the hill from the Foro Romano (Roman Forum), if entering from the east entrance, past the stairs the first major structure on the edge of the slope is the Teatro Del Fontanone, a fountain positioned between the short twin Uccellieri Towers. The terrace here is one of a few panoramic views on the Palantino overlooking the Forum.
Further on top of the hill are gardens and scattered shells of ancient elite residences. Some are fairly well preserved with tiles mosaics intact. There's an underground tunnel (Criptoportico Neroniano) lit up with multi-colored light displays showing how the interior may have appeared in Roman times, when royals used this as a connecting walkway between wings of the imperial palace.
Looking over the southern edge of Palantine Hill, the vast Circo Massimo is visible, almost the same length of the Forum, a grassy field that remains of the largest stadium dating back to several hundred years BCE.
On a weekend afternoon hundreds of tourists flocked to various points on the hill, though aside from the terraces it was only moderately crowded. More overhead tree cover is found here than below, which was nice since it was sunny and warm in April. Some metal placards are placed around the site with descriptions of archaeological features.
Access to Palatine Hill is via tickets to the Forum purchased on the Colosseum website (less expensive though harder to choose times), and can be bought in conjunction or separate from tickets for the Colosseum itself. I'd recommend 1-1.5 hours. Certainly worth checking out!