For home changes I'd avoid products designed to remove grease. They have chemicals that encourage rust. Best bet at home is to borrow the kitchen cheese grater and grate up some bar soap. Mix with warm water at about 50:50 and you have tyre soap. All you need to then is get your mucky finger marks off the cheese grater before wives/mothers/partners etc. spot 'em
If you used washing up liquid don't worry. I've used it plenty of times at the road side and it's fine, I just think that everytime I do i'm taking a few months off the life of the wheel.
The other trick with awkward beads is to inflate to the maximum cold pressure noted on the sidewall, then if it still won't pop on, bounce it round the carpark/drive. At home you can also leave the tyre in the sun (fat chance in UK today) or put it somewhere warm before you start. A warm tyre is a lot easier to handle. Some people suggest heating using paint stripper hot air guns, but they seem a bit fierce to me.
Andy