How can I make Dominion play better with 5+ players?
The biggest problem with extra players that I usually see is down-time. Dominion is partially a group solitaire game. There is some interaction between players with attack cards, but otherwise your opponents turns won't matter a lot besides keeping an idea on what piles are low and the game end. I find with 5 the main problem is "I know what I'm going to do, I just have to wait through 3 more people's turns to do it". You also might want to tweak the number of kingdom card piles to somewhere between 11 and 13.

The first and easiest thing you can do is lessen the cards that take longer to resolve. Spy, Throne room, Adventurer, Contraband, Counting House, Possession, and many others tend to make turns take a little longer. This doesn't mean that they are bad in any way, it just means if you are worried about long down-times with 5 or 6 people you might want to forgo them. Removing certain cards may annoy hardcore players who want a truly random layout, but my games usually aren't super serious and its worth it to fit in the extra person.

Another thing you can do if you are adventurous with 6 is play 3 teams of 2. Turns pass from one team to another. Both members of that team take their turns at the same time and have totally separate decks. Their attack cards only affect the players on the other teams. It helps to use attack cards that don't depend on order very much in case you both attack. What do you do if one of you plays a spy and the other a thief? What order do they go in? Instead of trying to make rules about this, just use attacks that don't clash with others too easily, like militia and witch. The team mechanism is mainly to allow some simultaneous turns; you can play as a team and do some minor collaboration, like who should buy what or who should focus on attacks vs. victory points, or still make it every man for themselves and have the player with the most victory points win.