Short answer: no, the CHL is not under Hockey Canada's umbrella, but they do cooperate as best they can.
The long story:
Hockey Canada is the organization under which most amateur hockey in Canada is played, and the organization which sits as the national member from Canada at the International Ice Hockey Federation (IIHF). Hockey Canada was founded in 1968 to run the national teams that played in IIHF tournaments and Olympic Games.
The Canadian Amateur Hockey Association (CAHA) was founded in 1914 to run amateur hockey in Canada. The CAHA had run the national teams prior to 1968: for international tournaments they used to send the Allan Cup (senior mens) champion teams as Canada's representatives. In '62 they stopped doing this and founded a national team program under the leadership of Father David Bauer (former Bruins great Bobby Bauer's little brother), who scouted players from the Canadian Intercollegiate Athletics Union (later Interuniversity, and in 2001 renamed CIS), which fell outside CAHA jurisdiction but was still considered amateur by the IIHF. The CAHA was itself an umbrella organization for the provincial hockey associations, which actually administered the organizational rules and ran the leagues of play. The CAHA's primary purpose was to set national standards for the provincial associations to follow.
All amateur leagues, from toddlers to adults, were run in accordance with the principles and rules agreed to by the provincial associations and enforced by the CAHA overall. That was until 1966, when Bill Hunter (owner of the Edmonton Oil Kings, then a Junior A team) convinced five members of the Saskatchewan Junior Hockey League to form a new Junior league outside the jurisdiction of the CAHA. The CAHA had been allowing the NHL teams to sponsor some of its Junior clubs, and that sponsorship of teams tended to be concentrated in Ontario and Quebec. Faced with competition for the Memorial Cup from these stacked teams from out east Hunter thought his team and some of the others from Saskatchewan would make a better go of it together, rather than playing in the smaller provincial leagues they had played in. (Hunter's Oil Kings were very good; they played in the Memorial Cup every year from '60 to '66, but only won twice in those seven years. He wanted to have the Oil Kings play in the Alberta Senior League rather than Junior, because the Oil Kings dominated their provincial Junior league. It was the CAHA's disallowing this that was the straw that broke the camel's back and convinced him to start his own interprovincial junior league.
Hunter's Canadian Major Junior Hockey League was given approval by the Alberta and Saskatchewan amateur hockey associations and began play in '66, without CAHA approval. The teams in the CMJHL were not allowed to compete for the Memorial Cup and were considered an "outlaw" league in the eyes of the CAHA. Kids who played in the CMJHL were not allowed to play in any CAHA-sanctioned events.
The CMJHL renamed itself the Western Canada Hockey League (WCHL) after the first season, and despite CAHA disapproval continued to play, and added teams from Manitoba.
In 1968 the Western Ontario Junior B Hockey League had approached its governing provincial association, the Ontario Hockey Association, for promotion to "Junior A" status. For many years it had been the best Junior B league in Ontario and wanted its teams to have a chance at playing for the Memorial Cup. The OHA refused, so the Western Ontario Junior B League went rogue and declared itself the Western Ontario Junior A League. They contacted the WCHL and proposed they play each other for their own national Junior A championship, outside the rules of the CAHA.
This rankled the CAHA, and some worry started to set in because these two "outlaw" leagues were getting pretty good. In particular the Flin Flon Bombers, one of the teams from Manitoba that joined the WCHL in 1968, was talked about as arguably the best junior team in the country. It didn't look good on the CAHA that the Bombers played outside their jurisdiction. Bobby Clarke, the Bombers' best player, was arguably the best junior player in the country.
Junior teams from Quebec saw how well the WCHL and WOJAHL were doing outside the auspices of the CAHA and in '69 teams from the Quebec Junior Hockey League, Metropolitan Montreal Junior Hockey League and a team from Ontario's Central Junior A Hockey League (the Cornwall Royals) joined forces to create the Quebec Major Junior Hockey League. The QMJHL stayed within the CAHA's jurisdiction but Junior A hockey was spiralling out of CAHA control pretty quickly.
In 1970 the CAHA relented amidst rumours the Ontario Hockey Association's Junior A league, the top Junior A league for decades, was going to go rogue too. They let the WCHL and WOJAHL back in the fold and they reorganized Junior A level hockey, splitting it in two: "Tier I" and "Tier II". The WCHL and QMJHL were considered "Tier I" leagues alongside the OHA's new Major Jr. A league; the WOJAHL, renamed the Southern Ontario Junior A Hockey League, fell back to "Tier II". The "Tier I" teams would compete for the Memorial Cup.
The OHA Major Jr. A league and QMJHL began a bitter feud in the midst of the '71 playoffs. The Memorial Cup had been the national championship of a very broad playoff structure, where provincial champion Junior A teams played each other in a series of games until western and eastern champions were crowned and faced each other for the Memorial Cup. When the CAHA reorganized Jr. A hockey in 1970 they didn't fundamentally change the format of the Memorial Cup: it was still east vs. west, except the WCHL champs got an automatic bye in while the OHA Major Jr. A and QMJHL champs played each other for the other berth. The OHA champion St. Catharines Black Hawks faced the Quebec Remparts for the Memorial Cup berth in '71 and the series became so acrimonious, so hostile, that the teams wouldn't play at the other's home rink after game four of the series. Game five was held at Maple Leaf Gardens. Games six and seven were scheduled for Quebec City but St. Catharines refused to go back to la Colisée and Quebec wouldn't agree to hold them in Montreal instead. St. Catharines withdrew entirely, and Quebec played Edmonton for the Memorial Cup by default.
For the '71-'72 season the CAHA changed the Memorial Cup into the round-robin tournament we know today (the hosts began getting an automatic bye in the '80s). In '72 the QMJHL petitioned the CAHA to refuse the Montreal Junior Canadiens' continued participation in the OHA, reasoning that they ought to play in Quebec (and the QMJHL was desperate to have a team in Montreal, the biggest city in the country at the time).
A far greater concern began overshadowing this acrimony in 1972 as well, but the full effects wouldn't be felt until 1974: the World Hockey Association. The CAHA had had a working agreement with the NHL to protect their Junior A clubs from having their rosters poached by pro clubs for a long time, going back to the '30s, but the WHA had no such agreement. The WHA began drafting, and signing, Junior players that had been on CAHA-sanctioned clubs without any compensation to the CAHA or to the individual clubs themselves. It got especially bad in '74, when the WHA began going after "underaged" (18-year-old) players in an effort to pre-empt the NHL from getting them. The WHA wasn't paying the Major Junior clubs for drafting players, it wasn't paying the CAHA, and the NHL wanted in on the "underaged" players too. The CAHA crafted an agreement with the NHL which allowed them to draft 18 and 19-year-olds and this ended up decimating the rosters of the Major Junior clubs.
In 1974 the OHA Major Junior A league decided to distance itself from the OHA, becoming the Ontario Major Junior Hockey League. The OMJHL was still nominally under control of the OHA but it had more independence; it crafted its own bylaws, it had its own commissioner, etc.
In 1975 the OMJHL, QMJHL and WCHL decided to create an overarching governing body for themselves: the Canadian Major Junior Hockey League (yes, the exact same name as Bill Hunter's original western league in '66). The CMJHL took over the negotiations with the NHL and WHA for draft pick compensation from the CAHA.
In 1980 the CMJHL split from the CAHA completely, reducing their relationship with the CAHA and its constituent provincial associations to a mere "affiliation". The CMJHL took complete control over "Tier I Junior A" hockey nationwide and changed it to "Major Junior" as we know it today. The CMJHL shortened its named to CHL in 1986. (The WCHL shortened its name to WHL in 1978, after it had grown to include teams in Seattle; Portland, Oregon and Billings, Montana. The OMJHL became OHL in 1980.) "Tier II Junior A" reverted to just "Junior A", and remained under CAHA control.
Hockey Canada had not only been created to sort out the national teams but was in fact meant to oversee all hockey in Canada: amateur, professional, collegiate. Everything. But the CAHA still had the actual de facto jurisdiction over the amateur game. Over the years the CAHA and Hockey Canada squabbled a bit and in 1994 Hockey Canada and the CAHA merged. By this time the CAHA had become very ineffectual and Hockey Canada had become embroiled in the scandal surrounding Alan Eagleson. (Eagle had been embezzling proceeds from the international tournaments he helped organize on Hockey Canada's behalf...) Hockey Canada is now effectively what the CAHA was pre-1968: it governs the amateur game from Junior A down and the Senior leagues, but is de facto powerless over the CHL and Major Junior Hockey. The CHL is considered a "partner" organization by Hockey Canada; they work together on committees and the like, but Hockey Canada doesn't have much say about how the CHL does business. In much the same way Hockey Canada "partners" with the NHL—NHL players are now allowed to play on the national teams, after all—but it has no jurisdiction over the NHL itself. This is also true of CIS.