|MadSci Network: Astronomy|
JPL has an excellent Halley's Comet orbit simulation Java Applet. It gives the location of the comet relative to the orbits of the planets. Using that Applet suggests the following:
Halley's comet reaches aphelion---its furthest distance from the Sun---at approximately 35.3 AU, during Christmas 2023. (1 AU is one Astronomical Unit, the average distance between the Sun and the Earth, equal to about 150 million kilometres).
Thirty-eight years later, in early 2061 February, it will once again reach perihelion, passing a mere 0.587 AU from the Sun, about midway between the orbits of Mercury and Venus. At that point Earth will be on the far side of the Sun from comet Halley, making the comet unviewable. However, observations should be good during Halley's approach and recession from the Sun, each side of perihelion.
In 2061 November the comet will reach a closest point 0.61 AU from the Earth, and will appear virtually overhead in the pre-midnight sky for viewers just north of the Equator. Once beyond perihelion, in 2062 April, about mid-month, Halley will again pass the Earth, reaching distances as close as 0.426 AU. At that point Halley will appear high overhead in the pre-dawn skies, best placed for observers in the southern hemisphere.
In terms of brightness, comets are notorious for their unpredictable displays, though it is likely that the situation in 2062 will at least match and possibly exceed the 1985 encounter.
Try the links in the MadSci Library for more information on Astronomy.