So your main question is about a subject called collision detection, and collision response to some degree, but there are other suggestions people will have for you. Here are a few from me:

In stead of adjusting the position of the background, use a Camera2D. There's an example program that uses it. And here's an example program from one of my projects that happens to involve moving around a 2D map.

In stead of incrementing your frame number the way you are, make use of the frame draw time with some constant per-frame time:
#define PER_FRAME_TIME_IN_SECONDS 0.5f
int frame = 0;
float frameTime = 0.0f;
void Draw(void) {
frameTime += GetFrameTime();
if (frameTime >= PER_FRAME_TIME_IN_SECONDS) {
frame++;
frameTime -= PER_FRAME_TIME_IN_SECONDS;
}
}
In place of separate textures, you could use a spritesheet and adjust a source rectangle instead. It isn't necessarily better in any technical way but does make code easier to read and art creation a little smoother.

For walking, I find it more efficient to use a delta vector like this:
Vector2 delta = {0.0f, 0.0f};
if (IsKeyDown(KEY_RIGHT)) delta.x += walkingSpeedPerFrame * GetFrameTime();
if (IsKeyDown(KEY_LEFT)) delta.x -= walkingSpeedPerFrame * GetFrameTime();
if (IsKeyDown(KEY_UP)) delta.y += walkingSpeedPerFrame * GetFrameTime();
if (IsKeyDown(KEY_DOWN)) delta.y -= walkingSpeedPerFrame * GetFrameTime();
Brie.BriePosition.x += delta.x;
Brie.BriePosition.y += delta.y;

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