First, I'll talk about my actual house rules, the hard-coded changes to the rules-as-written that I've made. Later I'll talk about the "softer" or subtler methodological things I do to make 5E work for me.
The house rules have evolved over time, but the set I have now is definitely one I'd use in the future. There are four categories: (1) XP (2) Inspiration (3) Resting (4) Stat Generation. I'll detail them below.
(I also wrote an abstract wealth / Resources system à la Burning Wheel for that White Plume Mountain / Against the Giants game of 5E, but that was out of necessity to that particular set of adventures, and isn't something I feel the need to do all the time. Basically, what happened is that my players and I all agreed that we didn't want to have to deal with the huge amount of arithmetic involved in those particular adventures.)
During sessions, gain 1 XP by:
Failing A Roll
(Exception: characters with multiple attacks in a round only gain XP for their attacks if they miss with all of them)
At end of session, gain 1 XP by:
Resolving and changing a BIFT
Discovering an interesting new fact about the setting
Acquiring a significant treasure
Forming a significant alliance
Making a powerful enemy who remains at large
Special End of Session Award: If the party overcomes a significant combat challenge, put XP into the pool equal to the difference between the party’s level and the CR of the most powerful monster defeated, with a minimum of 1 XP earned.
Each end of session award can only be earned once per session (Exception: each PC may resolve and change a BIFT). All XP goes into a pool, and the group levels up together when they reach the needed threshold of experience points. The XP total then resets to 0, unless there is a surplus, in which case the remainder stays in the pool. The party may level up during a Rest of any duration. (See the Resting rules, below.)
The actual chart of XP requirements can be found on the Google Doc
that this is all taken from. It's been carefully calibrated over time.