No. Xbox Live is a closed ecosystem for Xbox consoles. There is no official PC client that lets you sign into Xbox Live and play Xbox 360 games.
Some games support cross-platform play between PC and Xbox 360:
Note: Most Xbox 360 cross-play titles have retired servers.
Stream your PC gameplay to friends via Twitch/YouTube/Discord—they watch on Xbox 360 browser/app, but you're not playing together.
For true shared play, both parties need the same platform (or a modern cross-play title on current-gen hardware).