First of all, we need to clarify that there is no Mitsva in the Tora to perform the Kapparot, or any kind of animal sacriﬁce, in the eve of Yom Kippur. In our days, as Maimonides explains "en sham el-la teshuba", there is nothing else but Teshuba to atone for our sins. Teshuba is a serious mental and emotional process of retrospection, which consists in the admission and confession of our flaws and bad habits, which ultimately should lead us to improving our behavior .
Kapparot is a practice which was initiated by the common people -not by the rabbis- at the time of the Geonim (year 800-1000 CE). The Kappara made with a live animal suppose to help inspiring our Teshuba. By seeing the Shehita (slaughtering) of the chicken, we realize the extreme fragility of our lives. We then realize the appropriateness of doing Teshuba, reflecting on our own mortality. This is, by the way, the frequent explanation for the effect that the qorban, in the times of the Bet haMiqdash, had in the sinner's consciousness.
However, soon after the custom of Kapparot became more popular some prominent Rabbis like Ramban (Nahmanides) raised their voices against this practice. Maran Rabbi Yosef Caro (1488-1575) the author of the Shulhan Arukh, the supreme Code of Law for all Jews, disapproved the practice of Kapparot with chicken in the eve of Yom Kippur, explicitly and with very harsh terms. Moreover, in the first edition of the shulhan 'arukh (Venice 1565) it is written in the title of siman 605, where Rabbi Yosef Caro discuses Kapparot מנהג כפרות בערב יום כפור מנהג של שטות הוא. "The custom of Kapparot in the eve of Yom Kippur is a foolish custom". In later editions the editors erased the last line. (see the original edition of the Shulhan 'arukh here )
Why such opposition to the Kapparot?
First, as Nahmanides said: mishum darke haemori "because it is similar to the practice of idol worshipers" . (Even today, many cults like Macomb, Vodoo, etc. use a small chicken as a sacrifice to their deities. You can Google for example "Eshu" or "Elleggua" major idol-warriors of Santeria which must be worship by sacrificing to them a small chicken).
Second, the fact that the Kappara performance looks like the Korbanot performance, made many rabbis very concerned: because slaughtering an animal as a sacrifice outside the Bet haMiqdash (haqrabat hutz) is a serious Biblical transgression . For this reason many rabbis in the past forbade the consumption of the chicken that was slaughter in the fashion (or intention) of a qorban (The Rishba, while still opposed to the Kapparot with chicken, allowed to eat that chicken, which was usually given to the poor, because it is not one of the animals who were offered as a sacrifice in the Bet haMiqdash).
Third, since so many people want to do Kapparot on the eve of Yom Kippur, the Shehita might not be done with enough care and attention in terms of the checking of the the knives and other ritual details. This is the point brought by Rabbi Obadia Yosef z"l (who in the past supported the Kapparot) to favor the performance of Kapparot with Tsedaqa (see this).
There is yet another problem, which should be of a great concern when thinking about doing chicken Kapparot. Because of the great demand and the short time to slaughter so many animals, the chicken are too often mistreated in the process, left for days in cages without food or water. This is a serious transgression of an explicit Biblical prohibitions instructing us to treat animals with respect and avoid unnecessary suffering (tsa'ar ba'ale hayim). The new Ashkenazi Chief rabbi of Israel, Rabbi David Lau addressed this issue last year (see here ).
Lastly, if the person who does the Kappara would give directly the slaughtered chicken to a poor person, as it used to be done in the past, then we could view this Kappara as a form of Tsedaqa. But people who do the Kapparot would very often take seriously the idea of transference of his sins into the chicken and therefore not they, not even needy people would be eating those chicken "full of sins". Many of these chicken are not eaten but just wasted. And if one kills an animal unnecessarily (not to be eaten) one is transgressing another explicit Biblical prohibitions bal tashchit, i.e., unnecessary waste of the blessings, in this case an animal life, that God granted us.
In conclusion, the best way of doing Kapparot is by giving Tsedaqa (=charity for needy people). In this case, none of the above mentioned problems would apply, and in addition before Yom Kippur begins, you will be fulfilling the beautiful Biblical commandment of charity. Which is a Mitsva with no negative side effects. On the contrary, Tsedaqa was regarded as the most imortnat Mitvar 'ase, as Maimonides writes: "We have to care about the Mitsva of Tsedaqa more than [we care about] any other positive commandment" (MT, Matanot 'aniyim 10:1). Tsedaqa brings a great zekhut (merit) to the giver, and a great relief to the recipient וצדקה תציל ממות
See this video where rabbi Shelomo Aviner, Rosh Yeshibat Ateret Cohanim explains the ways of doing kapparot and mentions, among other things, that rabbi Kaddouri z"l and Rabbi Shlomo Zalman Auerbach z"l made kapparot with Tsedaqa instead of chicken.