The chart of recommended DCs by level in 4e has been done three times- once in the original DMG on the famous page 42, changed in the DMG2, and changed still further in the D&D Rules Compendium. Basically, the designers had a hard time nailing down the math for skills. And it’s easy to see why. We’ve seen how “to hit” bonuses in combat were basically kept uniform across various characters- the game assumes a +5 bonus at 1st level and increases from there through various means. However, the bonuses on individual skills can go all over the place- anywhere from -2 to +13. On top of all this, skill challenges, a noble attempt to provide some mechanical support to out of combat activities, were presented in an embryonic form and required some tweaking. There’s room for improvement, meaning the question is both how skills work in 4e and how they ought to work.
Let’s draw on what we know so far. The numbers for combat in D&D assume, for one, that a PC will hit a monster of their level 55% of a time- that is, on a roll of 10 or better. If we were to apply this to skills, what do we assume the PC’s bonus to be? It can range all over the place. A Level 1 character making a check when not trained in a skill and using their weakest ability score would have -2: at this low end, an appropriate difficulty of their level might be 8. (This may represent the “Easy” Level 1 DC, and sure enough in the Rules Compendium the DC for an easy level 1 check is 8.) A character with no proficiency and an average ability score would have no modifier, so they would make a DC 10 check 55% of the time. Someone with training but no ability bonus would have a 55% chance at a DC 15 check, etc.
Now, the Rules Compendium actually bases “Easy”, “Moderate”, and “Hard” DCs on, respectively, someone with no skill training and no modifier, positive or negative; someone with no training but an ability score of 18 or more; and someone with training and an 18+ ability. Throughout they favor a 65% success rate rather than a 55% one- the designers were leaning more in the direction of more hits (since a miss most often leads to nothing happening), and it could be argued that each skill roll carries a little more heft than an attack roll- you’re less likely to get another chance next turn if you miss. It’s less harsh than the DMG 1 (which seems to favor a 55% chance of success) but more challenging than the DMG2’s chart.
So really, doing skill DCs comes down to two things- what kind of PC can be expected to pull off an Easy/Moderate/Hard task of their level, and just what kind of probability “can be expected to” translates into. Mess with either of these inputs and you get a different chart, and if you want your charts legally distinct from anything published, you may have to do just that.
This is getting lengthy and I may have to split this up into multiple posts, but to summarize- skill DCs by level are based on how much a chance of success you want a PC of that level to have, and what bonuses they have. “Easy” DCs tend to assume someone with either no or negative bonuses, “Moderate” can go by either skill proficiency (+5) or a high ability score (anywhere from +3 to +5), and “Hard” assumes both. So it’s easy enough to make your own chart.