The first step is to look at the experience tables. The 5E standard party is assumed to be 5 players, just like the past few editions. You can multiply the experience chart figures x 5 to determine out how much cumulative experience a party needs to gain level 2, 3, and so on. It's simple to subtract the previous total to find out how much new experience the party needs each level. Example: a 5th level character (6,500 xp) needs 7,500xp to reach 14,000xp and 6th level. A party of five such characters would need to earn 37,500 to all advance from 5th to 6th level.
Next we turn our attention to the DMG's recommendations for encounter design and how much experience a party should earn for defeating an encounter. It's page 82 of the DMG (or page 56 of the free DM rules). Those pages demonstrate how to calculate experience budgets for encounters based on a difficulty range - easy, medium, hard, or deadly. In our case, I multiplied the values by 5 for the hard difficulty range to identify an experience budget per encounter level, and got the experience value for the minimum hard encounter for each level. From there, it's a simple formula to divide the amount of experience needed by the value of an encounter to see how many "hard" encounters a group needs to overcome in order to be ready to advance. All this is in the attached chart below.
In the comments to one of the recent megadungeon posts, a number of folks noted that advancement slows down quite a bit in the mid-levels, and sure enough, the # of "hard" encounters required shifts from 4, to 8, and stays at 10 for a big chunk of levels. I don't have much table-time with the 5E rules yet (okay, truthfully, I only did some playtests) but I'm assuming you can clear about 4 encounters in a 4 hour dungeon crawl. That would let you clear levels 2 and 3 in a single game session, get to level 4 after 2 more sessions, and require 2-3 sessions for each level past 4. That's right in line with the pacing suggested on "Level Advancement Without XP" on page 261. It will take you 36-37 sessions of straight encounters to get to level 20 - less than a year if you play once a week.
Caveats apply: I only used an average value of "hard" encounters, whereas a real campaign will have a blend of easy, medium, hard, and deadly encounters to throw at the party. Maybe the average to clear level 1 is more like 5-6 sundry encounters, and not just 4 hard encounters. It looks like the scant published adventures for 5E supplement encounter experience with some quest or milestone based awards, too.
Hope this was interesting! I'm sure I'll discover additional ramifications of the numbers, but my first goal was just to understand how much content players need to encounter at each level before being ready to move on. For instance, there's not much point in preparing 30 level-1 combat encounters in an area if the party is going to be ready to move onto bigger and better things after encountering only 4 of them. How much content to prepare? That's a question for another day - perhaps tomorrow.