If someone empties / changes his/her colonostomy bag, should s/he say "Asher Yatzar"? I would assume, yes, as this is the person's method of ridding waste. I assume the same answer would apply to someone doing dialysis? Or, is the bracha limited to only the "standard way" of "going to the bathroom"?
One with a colonostomy generally has 3 excretions per day and should make the bracha of Asher Yatzar after each excretion. Nishmas Avraham (ed 2) Orach Chaim 7:1;note 7)
Rav SY Elyashiv, Rav SZ Auerbach and Tzitz Eliezer write that he can only make the bracha after cleaning and covering the area. Nishmas Avraham (ed 2) Orach Chaim 76:9;note 7)
See pg 24, fn 230 here for an English discussion by Rabbi Dr Avraham Steinberg.
R. Tzvi P. Frank (Har Zvi OC 6) was asked about one who has to relieve himself via a catheter and ruled that Asher Yazar is recited because there still exists the possibility of a malfunction and "disconnect" in which case the person would be left in pain. This opinion appears to be the consensus among contemporary poskim (cf. Birchat Eitan, Bnei Brak 2007, pg. 445). Now, although R. Frank may have been discussing a case where the catheter was inserted via the urethra, one may not be amiss in applying his reasoning to both dialysis and colostomy as well. Indeed R. Waldenberg (Tzitz Eliezer vol. 8 no. 1, end) applies the same to one who has a "catheter running into the stomach" (colostomy?) or via the kidney that he does recite the blessing. However, students of R. Shlomo Z. Auerbach record in his name (Halichos Shlomo, ch. 20, sec. Dvar Halacha n. 43) that while one on dialysis is obligated to thank God for the successful procedure he does not recite Asher Yazar since the blessing was not instituted where the body cleanses itself in this manner. R. Yitzchok Liebes (Beis 'Avi vol. 2 no. 22), after considering the same argument proposed by R. Auerbach, ruled that one should still recite the blessing even though the cleansing is not done the natural way; so ruled R. Isaac Herzog too (Heichal Yitzchok OC 4).
Re. the time to recite the blessing and what to do with the receptacle, the consensus from the above sources appear to rule that if there is a continuous drip, or one is not able to realize when any discharge occurred they should recite the blessing once in the morning and have in mind for the rest of the day. If, however, they can know or feel every time they were relieved they should recite it each time. The receptacle may be present as long as it is enclosed and does not emit a foul odor. If it contains feces or is in a clear tube it should be covered.