If you want a package from the UK, then probably a company like Rocketski or similar is your most budget option, not sure you'd get everything like a pool, but to get a February week, you are going to have to book well in advance or keep fingers crossed for a last minute bargain, very hard to find in February.
You will almost definitely be able to do a cheaper holiday if you make your own package. The French prefer "compact" accommodation, to them the skiing and location is more important, they got for self catering and using the bread collection service for breakfast (nearly all apartments have this), then either eat in the apartment restaurant or other restaurant nearby, or cook for yourselves. Many French apartment buildings include pools, many are right next to the slopes, the lifts and/or ski schools. French apartment companies are Pierre et Vacances, MGM, Goelia, Odalys, Maeva, Goelia, Residhotel, that's just a few names, there are many more of them. I'd reckon Goelia are possibly the cheapest.
There are resorts which give discounts for families/groups buying lift passes together (don't have to be blood related at some).
Ski hire is often cheapest by booking ahead by internet - always try to get a hire shop near to where you are staying which is open late the day before skiing or very early the next morning, if you are doing lessons, so you are not late.
You could fly with a discount airline such as Easyjet to Geneva and access many northern Alps resorts.
I started off research for our first skiing holiday by using the Peak Retreats website. We've only actuallly booked once for them, I just really appreciate the information on their website.
I personally don't think Bulgaria is particularly cheap, maybe when comparing UK travel operators, but comparing a home made package, France or Austria would be often cheaper.
Do consider the fact that in 2012, Easter falls on 9th April. French schools don't go on holiday until the week after, whereas most UK schools are on holiday the week before Easter, so the first week of Easter might be a good week to go rather than February, just choose a more snow secure resort, the prices in April are always alot cheaper than February.