If your coming from Vietnam, I assume the your first city will be Kunming- you might want to check for Tibet ‘budget’ deals there (There’s a guy at the Camellia, I think). This should include a flight, other travellers for a so-called ‘group’, and probably a taxi from the airport and a few days accommodation in a place you wont like.
Doing the same in Chengdu will probably be a bit cheaper- 1600-2000 yaun (200$US-250$US), though its a 20h train there, hard sleeper is about 230 yuan. But then again, you will get to see (and taste) Sichuan.
Your other legal options include flying from Lijiang (northwestern Yunnan, recommended by itself), or taking an overland bus from Golmud, past Xining- thats north of Tibet, and would take at least 3 days to get there. The price not being much cheaper (1300-1600, as I understand), there’s no real reason to do it.

On top of these, you have 3 highways climbing Tibet from that side- the Yunnan-Tibet HWY, from Deqin, at the north western tip of Yunnan, would be a wonderful way to get into Tibet, as would the northern and southern HWYs from Sichuan. The problem? It ain’t strictly legal, you have to hitch and hick, a hole lot of luck, and might end up fined and sent back by the police.
That in mind, I did meet at least 5 guys who made it onto Tibet- some hitching and some cycling. An Asian appearance certainly helps, and so does basic knowledge of Chinese.

As for the weather- I was there early October (after monsoon season), and had great weather. April would probably be colder, but not a bad time to go.
Check before you buy the ticket ;)