Well, you want to do a few different things during & after the move.
These are pretty general, since I don't know your exact situation.
1. Make sure the account looks built-up. At least 5 years old if not more. You want to make sure it looks well lived-in, not just an empty shell, because that's suspicious.
2. Just don't go on the Pound Chat. I promise you, it's not worth the risk when people are report-happy as they are over even common, poorly-named UCs.
3. Get a back-story. Maybe you had an old side account that you made a long time ago and forgot about, and you found a UC on there, and you're moving it up. If that account has other cool pets on it, maybe you traded. Have answers ready if you get asked questions.
4. If the UC's something particularly valuable or well-named, do some research first to see if there's anybody out there keeping an eye on the specific pet. The higher up you go on the tier ladder, the more likely it is that you have someone lurking that pet.
5. Probably put some more pets on that account. Just one pet is... odd, at least to me-- it's hard to have just one. Or two, in this case. Maybe some VWN + customized randoms with decent colors. Maybe one with good stats. Y'know, make your account look lived-in.
6. "How" to move? Depends on what's already on there. If you've got a RW or other valuable pet somewhere else, you could move that to this account, then trade that for the UC in question. Otherwise, have the accounts exchange neomails to show a realistic conversation about the transfer. Make it look real.
6b. This is assuming it's a bought UC. If it's a legitimate UC, then I feel that you have a few less things to worry about, but do still be careful and try to avoid getting too much attention.