When I start from the beginning, I don't seem to ever get any - am I just getting unlucky? Does their drop rate change from the beginning of the act to the end?
The chances of getting an item of each item level are even throughout an entire Act. The hotfix note provides "approximate" rates because there are minor variances based on item type, and the % listed is an approximate aggregate across all item types (for the item level in a specific Act).
In an effort to make the answer more general, I'll include the updated drop rates for all zones. From the patch notes posted here, the approximate drop rates for all zones are:
Hell - Act III and Act IV
- iLvl 61: 9%
- iLvl 62: 1.9%
- iLvl 63: 0%
Inferno - Act I
- iLvl 61: 17.7%
- iLvl 62: 7.9%
- iLvl 63: 2.0% (This answers your specific question)
Inferno - Act II
- iLvl 61: 18.6%
- iLvl 62: 12.4%
- iLvl 63: 4.1%
Inferno - Act III and Act IV
- iLvl 61: 24.1%
- iLvl 62: 16.1%
- iLvl 63: 8.0%
Given a 2% drop rate for Act I, you are probably just unlucky - or rather, you'd be very lucky if you got one.
The entirety of Act 1 Inferno has a 2% chance of dropping items with an ilvl of 63. In fact, the drop rate of items with ilvls of 61, 62, and 63 are constant through the whole of each act.
See the patch notes for more information.
From my experience, Act 1 inferno can drop item level 63 from the start to the end of the act. I have received 11 item level 63 items since the change, playing the entire act a few times.
I've gotten one off of the trash right in front of New Tristram, before even going in the town. I've gotten a few from the quest's that come before the blacksmith. I've also gotten a few off of the Halls of Agony.
Note, I've never gotten one from the Warden or the Butcher, but I'm pretty sure that's just luck/bad luck. All except for one has come from champ/elite packs (most recent one came off of a white mob in front of New Tristram).
After the patch 1.04 update i went through all act I inferno 2 times with a solo demon hunter + mf/gf follower (templar), warded 450k gold per run (101% gf, 156% mf) and got some AH sales, making 1,2kk gold at all. I have droped a good yellow bow level 63 with 1090 dps + DEX + socket, in a elite pack on cemetery that turned into my primary weapon and the best bow i had ever droped. Also got some decent hand crossbows >770 dps with good stats and some armor level 61-63 that turned into my primary armor. It was very profitable. Only recoreded 2 deaths for some elite pack combinations. However that were no legendaries. Each run got about 4 hours.