Does Halley's Comet travel past the outer bounds of the Oort Cloud? We know Halley's Comet returns every 75-76 years. We can reasonably compute its elliptical orbit.
We know that the Oort Cloud is a cloud of predominantly icy planetesimals proposed to surround the Sun at distances ranging from 2,000 to 200,000 au. We divide it into  two regions: a disc-shaped inner Oort cloud (or Hills cloud) and a spherical outer Oort cloud.
We know that some comets are thought to have originated from the Oort cloud.
My question is: Does Halley's Comet travel past the outer bounds of the Oort Cloud?
No. Halley's Comet has an aphelion of 35 AU, which is  far less than the believed boundaries of the Oort cloud.
Any object on an elliptical orbit  with a perihelion of 0.6 AU and an aphelion of 2,000 AU has a semimajor axis of about 1,000 AU, and orbital period of about 31,600 years, much longer than Halley's Comet's current orbital period of 75-76 years. To get out to 200,000 AU and back on an elliptical orbit, you'd be looking at an orbital period of more than 31.6 million years.
As mentioned by @NilayGhosh and @planetmaker, 1P/Halley likely started as an Oort Clout object, was perturbed into becoming a long-period comet, and probably  subsequently became the prototypical periodic comet through gravitational interactions with the solar system's gas giants at some point in the last couple hundred thousand years.
As such, though it is possible that Halley's comet may have been in an orbit whose aphelion went beyond the outer boundary of the Oort Cloud in its long-period-comet days, it certainly does not do so now.